Theorem ax11a2 1809

Description: Derive ax-11o 1811 from a hypothesis in the form of ax-11 1494. The hypothesis is even weaker than ax-11 1494, 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 1810. (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 1514	. . 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 1808	1 |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1341

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751

This theorem is referenced by:  ax11o  1810