Theorem ax11o 1802

Description: Derivation of set.mm's original ax-11o 1803 from the shorter ax-11 1486 that has replaced it.

An open problem is whether this theorem can be proved without relying on ax-16 1794 or ax-17 1506.

Normally, ax11o 1802 should be used rather than ax-11o 1803, 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 1486	. 2 |- ( x = z -> ( A. z ph -> A. x ( x = z -> ph ) ) )
2	1	ax11a2 1801	1 |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743

This theorem is referenced by:  ax11b  1806  equs5  1809