Theorem ax11a2 1814

Description: Derive ax-11o 1816 from a hypothesis in the form of ax-11 1499. The hypothesis is even weaker than ax-11 1499, with z both distinct from x and not occurring in ph. Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1815. (Contributed by NM, 2-Feb-2007.)

Hypothesis

Ref	Expression
ax11a2.1	|- ( x = z -> ( A. z ph -> A. x ( x = z -> ph ) ) )

Assertion

Ref	Expression
ax11a2	|- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )

Distinct variable groups:   x, z   y, z   ph, z

Allowed substitution hints:   ph( x, y)

Proof of Theorem ax11a2

Step	Hyp	Ref	Expression
1		ax-17 1519	. . 3 |- ( ph -> A. z ph )
2		ax11a2.1	. . 3 |- ( x = z -> ( A. z ph -> A. x ( x = z -> ph ) ) )
3	1, 2	syl5 32	. 2 |- ( x = z -> ( ph -> A. x ( x = z -> ph ) ) )
4	3	ax11v2 1813	1 |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1346

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 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756

This theorem is referenced by:  ax11o  1815