Theorem ax11a2 1742

Description: Derive ax-11o 1744 from a hypothesis in the form of ax-11 1437. The hypothesis is even weaker than ax-11 1437, 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 1743. (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 1459	. . 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 1741	1 |- ( -. A. x x = y -> ( x = y -> ( ph -> A. x ( x = y -> ph ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1282

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by:  ax11o  1743