Theorem ax11o 1700

Description: Derivation of set.mm's original ax-11o 1701 from the shorter ax-11 1394 that has replaced it.

An open problem is whether this theorem can be proved without relying on ax-16 1692 or ax-17 1416.

Normally, ax11o 1700 should be used rather than ax-11o 1701, 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 1394	. 2 ⊢ (x = z → (∀zφ → ∀x(x = z → φ)))
2	1	ax11a2 1699	1 ⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1240

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 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643

This theorem is referenced by:  ax11b  1704  equs5  1707