Theorem equsalh 1611

Description: A useful equivalence related to substitution. New proofs should use equsal 1612 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.xps)
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.xps)
3	2	19.3h 1442	. . . . 5 |- (A.xps <-> ps)
4	1, 3	syl6bbr 187	. . . 4 |- (x = y -> (ph <-> A.xps))
5	4	pm5.74i 169	. . 3 |- ((x = y -> ph) <-> (x = y -> A.xps))
6	5	albii 1356	. 2 |- (A.x(x = y -> ph) <-> A.x(x = y -> A.xps))
7	2	a1d 22	. . . 4 |- (ps -> (x = y -> A.xps))
8	2, 7	alrimih 1355	. . 3 |- (ps -> A.x(x = y -> A.xps))
9		ax9o 1585	. . 3 |- (A.x(x = y -> A.xps) -> ps)
10	8, 9	impbii 117	. 2 |- (ps <-> A.x(x = y -> A.xps))
11	6, 10	bitr4i 176	1 |- (A.x(x = y -> ph) <-> ps)

Syntax hints:   -> wi 4   <-> wb 98  A.wal 1240

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  sb6x  1659  dvelimfALT2  1695  dvelimALT  1883  dvelimfv  1884  dvelimor  1891