Theorem equsalh 1726

Description: A useful equivalence related to substitution. New proofs should use equsal 1727 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)

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.2	. . . . 5 |- ( x = y -> ( ph <-> ps ) )
2		equsalh.1	. . . . . 6 |- ( ps -> A. x ps )
3	2	19.3h 1553	. . . . 5 |- ( A. x ps <-> ps )
4	1, 3	bitr4di 198	. . . 4 |- ( x = y -> ( ph <-> A. x ps ) )
5	4	pm5.74i 180	. . 3 |- ( ( x = y -> ph ) <-> ( x = y -> A. x ps ) )
6	5	albii 1470	. 2 |- ( A. x ( x = y -> ph ) <-> A. x ( x = y -> A. x ps ) )
7	2	a1d 22	. . . 4 |- ( ps -> ( x = y -> A. x ps ) )
8	2, 7	alrimih 1469	. . 3 |- ( ps -> A. x ( x = y -> A. x ps ) )
9		ax9o 1698	. . 3 |- ( A. x ( x = y -> A. x ps ) -> ps )
10	8, 9	impbii 126	. 2 |- ( ps <-> A. x ( x = y -> A. x ps ) )
11	6, 10	bitr4i 187	1 |- ( A. x ( x = y -> ph ) <-> ps )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1351

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-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-i9 1530  ax-ial 1534

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  sb6x  1779  dvelimfALT2  1817  dvelimALT  2010  dvelimfv  2011  dvelimor  2018