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Known
df_nfOLD__ax_gen__ax_4__ax_5__df_sb__df_sb_b__ax_6__ax_7__ax_7_b__ax_7_b1__ax_8__ax_8_b__ax_8_b1__ax_9__ax_9_b__ax_9_b1__ax_10__ax_11
:
∀ x0 : ο .
(
(
∀ x1 :
ι → ο
.
wb
(
wnfOLD
x1
)
(
∀ x2 .
x1
x2
⟶
∀ x3 .
x1
x3
)
)
⟶
(
∀ x1 :
ι → ο
.
(
∀ x2 .
x1
x2
)
⟶
∀ x2 .
x1
x2
)
⟶
(
∀ x1 x2 :
ι → ο
.
(
∀ x3 .
x1
x3
⟶
x2
x3
)
⟶
(
∀ x3 .
x1
x3
)
⟶
∀ x3 .
x2
x3
)
⟶
(
∀ x1 : ο .
x1
⟶
∀ x2 .
x1
)
⟶
(
∀ x1 :
ι → ο
.
∀ x2 x3 .
wb
(
wsb
x1
x2
)
(
wa
(
wceq
(
cv
x3
)
(
cv
x2
)
⟶
x1
x3
)
(
wex
(
λ x4 .
wa
(
wceq
(
cv
x4
)
(
cv
x2
)
)
(
x1
x4
)
)
)
)
)
⟶
(
∀ x1 :
ι → ο
.
∀ x2 .
wb
(
wsb
x1
x2
)
(
wa
(
wceq
(
cv
x2
)
(
cv
x2
)
⟶
x1
x2
)
(
wex
(
λ x3 .
wa
(
wceq
(
cv
x3
)
(
cv
x3
)
)
(
x1
x3
)
)
)
)
)
⟶
(
∀ x1 .
wn
(
∀ x2 .
wn
(
wceq
(
cv
x2
)
(
cv
x1
)
)
)
)
⟶
(
∀ x1 x2 x3 .
wceq
(
cv
x1
)
(
cv
x2
)
⟶
wceq
(
cv
x1
)
(
cv
x3
)
⟶
wceq
(
cv
x2
)
(
cv
x3
)
)
⟶
(
∀ x1 x2 .
wceq
(
cv
x1
)
(
cv
x2
)
⟶
wceq
(
cv
x1
)
(
cv
x1
)
⟶
wceq
(
cv
x2
)
(
cv
x1
)
)
⟶
(
∀ x1 x2 .
wceq
(
cv
x1
)
(
cv
x2
)
⟶
wceq
(
cv
x1
)
(
cv
x2
)
⟶
wceq
(
cv
x2
)
(
cv
x2
)
)
⟶
(
∀ x1 x2 x3 .
wceq
(
cv
x1
)
(
cv
x2
)
⟶
wcel
(
cv
x1
)
(
cv
x3
)
⟶
wcel
(
cv
x2
)
(
cv
x3
)
)
⟶
(
∀ x1 x2 .
wceq
(
cv
x1
)
(
cv
x2
)
⟶
wcel
(
cv
x1
)
(
cv
x1
)
⟶
wcel
(
cv
x2
)
(
cv
x1
)
)
⟶
(
∀ x1 x2 .
wceq
(
cv
x1
)
(
cv
x2
)
⟶
wcel
(
cv
x1
)
(
cv
x2
)
⟶
wcel
(
cv
x2
)
(
cv
x2
)
)
⟶
(
∀ x1 x2 x3 .
wceq
(
cv
x1
)
(
cv
x2
)
⟶
wcel
(
cv
x3
)
(
cv
x1
)
⟶
wcel
(
cv
x3
)
(
cv
x2
)
)
⟶
(
∀ x1 x2 .
wceq
(
cv
x1
)
(
cv
x2
)
⟶
wcel
(
cv
x1
)
(
cv
x1
)
⟶
wcel
(
cv
x1
)
(
cv
x2
)
)
⟶
(
∀ x1 x2 .
wceq
(
cv
x1
)
(
cv
x2
)
⟶
wcel
(
cv
x2
)
(
cv
x1
)
⟶
wcel
(
cv
x2
)
(
cv
x2
)
)
⟶
(
∀ x1 :
ι → ο
.
wn
(
∀ x2 .
x1
x2
)
⟶
∀ x2 .
wn
(
∀ x3 .
x1
x3
)
)
⟶
(
∀ x1 :
ι →
ι → ο
.
(
∀ x2 x3 .
x1
x2
x3
)
⟶
∀ x2 x3 .
x1
x3
x2
)
⟶
x0
)
⟶
x0
Theorem
df_nfOLD
:
∀ x0 :
ι → ο
.
wb
(
wnfOLD
x0
)
(
∀ x1 .
x0
x1
⟶
∀ x2 .
x0
x2
)
(proof)
Theorem
ax_gen
:
∀ x0 :
ι → ο
.
(
∀ x1 .
x0
x1
)
⟶
∀ x1 .
x0
x1
(proof)
Theorem
ax_4
:
∀ x0 x1 :
ι → ο
.
(
∀ x2 .
x0
x2
⟶
x1
x2
)
⟶
(
∀ x2 .
x0
x2
)
⟶
∀ x2 .
x1
x2
(proof)
Theorem
ax_5
:
∀ x0 : ο .
x0
⟶
∀ x1 .
x0
(proof)
Theorem
df_sb
:
∀ x0 :
ι → ο
.
∀ x1 x2 .
wb
(
wsb
x0
x1
)
(
wa
(
wceq
(
cv
x2
)
(
cv
x1
)
⟶
x0
x2
)
(
wex
(
λ x3 .
wa
(
wceq
(
cv
x3
)
(
cv
x1
)
)
(
x0
x3
)
)
)
)
(proof)
Theorem
df_sb_b
:
∀ x0 :
ι → ο
.
∀ x1 .
wb
(
wsb
x0
x1
)
(
wa
(
wceq
(
cv
x1
)
(
cv
x1
)
⟶
x0
x1
)
(
wex
(
λ x2 .
wa
(
wceq
(
cv
x2
)
(
cv
x2
)
)
(
x0
x2
)
)
)
)
(proof)
Theorem
ax_9d2
:
∀ x0 .
wn
(
∀ x1 .
wn
(
wceq
(
cv
x1
)
(
cv
x0
)
)
)
(proof)
Theorem
ax_8d
:
∀ x0 x1 x2 .
wceq
(
cv
x0
)
(
cv
x1
)
⟶
wceq
(
cv
x0
)
(
cv
x2
)
⟶
wceq
(
cv
x1
)
(
cv
x2
)
(proof)
Theorem
ax_7_b
:
∀ x0 x1 .
wceq
(
cv
x0
)
(
cv
x1
)
⟶
wceq
(
cv
x0
)
(
cv
x0
)
⟶
wceq
(
cv
x1
)
(
cv
x0
)
(proof)
Theorem
ax_7_b1
:
∀ x0 x1 .
wceq
(
cv
x0
)
(
cv
x1
)
⟶
wceq
(
cv
x0
)
(
cv
x1
)
⟶
wceq
(
cv
x1
)
(
cv
x1
)
(proof)
Theorem
ax_8
:
∀ x0 x1 x2 .
wceq
(
cv
x0
)
(
cv
x1
)
⟶
wcel
(
cv
x0
)
(
cv
x2
)
⟶
wcel
(
cv
x1
)
(
cv
x2
)
(proof)
Theorem
ax_8_b
:
∀ x0 x1 .
wceq
(
cv
x0
)
(
cv
x1
)
⟶
wcel
(
cv
x0
)
(
cv
x0
)
⟶
wcel
(
cv
x1
)
(
cv
x0
)
(proof)
Theorem
ax_8_b1
:
∀ x0 x1 .
wceq
(
cv
x0
)
(
cv
x1
)
⟶
wcel
(
cv
x0
)
(
cv
x1
)
⟶
wcel
(
cv
x1
)
(
cv
x1
)
(proof)
Theorem
ax_9
:
∀ x0 x1 x2 .
wceq
(
cv
x0
)
(
cv
x1
)
⟶
wcel
(
cv
x2
)
(
cv
x0
)
⟶
wcel
(
cv
x2
)
(
cv
x1
)
(proof)
Theorem
ax_9_b
:
∀ x0 x1 .
wceq
(
cv
x0
)
(
cv
x1
)
⟶
wcel
(
cv
x0
)
(
cv
x0
)
⟶
wcel
(
cv
x0
)
(
cv
x1
)
(proof)
Theorem
ax_9_b1
:
∀ x0 x1 .
wceq
(
cv
x0
)
(
cv
x1
)
⟶
wcel
(
cv
x1
)
(
cv
x0
)
⟶
wcel
(
cv
x1
)
(
cv
x1
)
(proof)
Theorem
ax_10
:
∀ x0 :
ι → ο
.
wn
(
∀ x1 .
x0
x1
)
⟶
∀ x1 .
wn
(
∀ x2 .
x0
x2
)
(proof)
Theorem
ax_wl_11v
:
∀ x0 :
ι →
ι → ο
.
(
∀ x1 x2 .
x0
x1
x2
)
⟶
∀ x1 x2 .
x0
x2
x1
(proof)