Theorem ax11o 1681

Description: Derivation of set.mm's original ax-11o 1682 from the shorter ax-11 1374 that has replaced it.

An open problem is whether this theorem can be proved without relying on ax-16 1673 or ax-17 1396.

Normally, ax11o 1681 should be used rather than ax-11o 1682, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.)

Assertion

Ref	Expression
ax11o	⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))

Proof of Theorem ax11o

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-11 1374	. 2 ⊢ (x = z → (∀zφ → ∀x(x = z → φ)))
2	1	ax11a2 1680	1 ⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1224

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405

This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624

This theorem is referenced by:  ax11b  1685  equs5  1688