Theorem a16g 1792

Description: A generalization of axiom ax-16 1742. (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 1740	. 2 |- ( A. x x = y -> A. z z = x )
2		ax16 1741	. 2 |- ( A. x x = y -> ( ph -> A. x ph ) )
3		biidd 170	. . . 4 |- ( A. z z = x -> ( ph <-> ph ) )
4	3	dral1 1665	. . 3 |- ( A. z z = x -> ( A. z ph <-> A. x ph ) )
5	4	biimprd 156	. 2 |- ( A. z z = x -> ( A. x ph -> A. z ph ) )
6	1, 2, 5	sylsyld 57	1 |- ( A. x x = y -> ( ph -> A. z ph ) )

Syntax hints:   -> wi 4   A.wal 1287

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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693

This theorem is referenced by:  a16gb  1793  a16nf  1794