Theorem equsalh 2139

Description: An equivalence related to implicit substitution. (Contributed by NM, 2-Jun-1993.)

Hypotheses

Ref	Expression
equsalh.1	|- ( ps -> A. x ps )
equsalh.2	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
equsalh	|- ( A. x ( x = y -> ph ) <-> ps )

Proof of Theorem equsalh

Step	Hyp	Ref	Expression
1		equsalh.1	. . 3 |- ( ps -> A. x ps )
2	1	nfi 1684	. 2 |- F/ x ps
3		equsalh.2	. 2 |- ( x = y -> ( ph <-> ps ) )
4	2, 3	equsal 2138	1 |- ( A. x ( x = y -> ph ) <-> ps )

Syntax hints:   -> wi 4   <-> wb 189   A.wal 1452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943  ax-13 2101

This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-nf 1678

This theorem is referenced by:  dvelimf-o  32544