Theorem bj-equsal 32813

Description: Shorter proof of equsal 2291. (Contributed by BJ, 30-Sep-2018.) Proof modification is discouraged to avoid using equsal 2291, but "min */exc equsal" is ok. (Proof modification is discouraged.)

Hypotheses

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

Assertion

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

Proof of Theorem bj-equsal

Step	Hyp	Ref	Expression
1		bj-equsal.1	. . 3 |- F/ x ps
2		bj-equsal.2	. . . 4 |- ( x = y -> ( ph <-> ps ) )
3	2	biimpd 219	. . 3 |- ( x = y -> ( ph -> ps ) )
4	1, 3	bj-equsal1 32811	. 2 |- ( A. x ( x = y -> ph ) -> ps )
5	2	biimprd 238	. . 3 |- ( x = y -> ( ps -> ph ) )
6	1, 5	bj-equsal2 32812	. 2 |- ( ps -> A. x ( x = y -> ph ) )
7	4, 6	impbii 199	1 |- ( A. x ( x = y -> ph ) <-> ps )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   F/wnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)