Theorem equsal 1615

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 1568	. 2 |- ( A. x ( x = y -> ps ) <-> ( E. x x = y -> ps ) )
3		equsal.2	. . . 4 |- ( x = y -> ( ph <-> ps ) )
4	3	pm5.74i 169	. . 3 |- ( ( x = y -> ph ) <-> ( x = y -> ps ) )
5	4	albii 1359	. 2 |- ( A. x ( x = y -> ph ) <-> A. x ( x = y -> ps ) )
6		a9e 1586	. . 3 |- E. x x = y
7	6	a1bi 232	. 2 |- ( ps <-> ( E. x x = y -> ps ) )
8	2, 5, 7	3bitr4i 201	1 |- ( A. x ( x = y -> ph ) <-> ps )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   F/wnf 1349   E.wex 1381

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 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by:  intirr  4711  fun11  4966