Theorem equsal 1715

Description: A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 5-Feb-2018.)

Hypotheses

Ref	Expression
equsal.1	|- F/ x ps
equsal.2	|- ( x = y -> ( ph <-> ps ) )

Assertion

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

Proof of Theorem equsal

Step	Hyp	Ref	Expression
1		equsal.1	. . 3 |- F/ x ps
2	1	19.23 1666	. 2 |- ( A. x ( x = y -> ps ) <-> ( E. x x = y -> ps ) )
3		equsal.2	. . . 4 |- ( x = y -> ( ph <-> ps ) )
4	3	pm5.74i 179	. . 3 |- ( ( x = y -> ph ) <-> ( x = y -> ps ) )
5	4	albii 1458	. 2 |- ( A. x ( x = y -> ph ) <-> A. x ( x = y -> ps ) )
6		a9e 1684	. . 3 |- E. x x = y
7	6	a1bi 242	. 2 |- ( ps <-> ( E. x x = y -> ps ) )
8	2, 5, 7	3bitr4i 211	1 |- ( A. x ( x = y -> ph ) <-> ps )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   F/wnf 1448   E.wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by:  intirr  4990  fun11  5255