Theorem a16g 1886

Description: A generalization of Axiom ax-16 1836. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
a16g	|- ( A. x x = y -> ( ph -> A. z ph ) )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem a16g

Step	Hyp	Ref	Expression
1		aev 1834	. 2 |- ( A. x x = y -> A. z z = x )
2		ax16 1835	. 2 |- ( A. x x = y -> ( ph -> A. x ph ) )
3		biidd 172	. . . 4 |- ( A. z z = x -> ( ph <-> ph ) )
4	3	dral1 1752	. . 3 |- ( A. z z = x -> ( A. z ph <-> A. x ph ) )
5	4	biimprd 158	. 2 |- ( A. z z = x -> ( A. x ph -> A. z ph ) )
6	1, 2, 5	sylsyld 58	1 |- ( A. x x = y -> ( ph -> A. z ph ) )

Syntax hints:   -> wi 4   A.wal 1370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785

This theorem is referenced by:  a16gb  1887  a16nf  1888