Theorem ax11o 1610

Description: Derivation of set.mm's original ax-11o 1611 from the shorter ax-11 1331 that has replaced it.

An open problem is whether this theorem can be proved without relying on ax-16 1602 or ax-17 1352.

Normally, ax11o 1610 should be used rather than ax-11o 1611, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.)

Assertion

Ref	Expression
ax11o	|- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))

Proof of Theorem ax11o

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-11 1331	. 2 |- (x = z -> (A.zph -> A.x(x = z -> ph)))
2	1	ax11a2 1609	1 |- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))

Syntax hints:  -. wn 3   -> wi 4  A.wal 1267

This theorem is referenced by:  equs5  1617

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 97  ax-ia2 98  ax-ia3 99  ax-in2 528  ax-io 609  ax-5 1268  ax-7 1269  ax-gen 1270  ax-ie1 1315  ax-ie2 1316  ax-8 1329  ax-10 1330  ax-11 1331  ax-i12 1332  ax-4 1334  ax-17 1352  ax-i9 1356  ax-ial 1361

This theorem depends on definitions:  df-bi 108  df-nf 1282  df-sb 1554