Theorem equsalv 2108

Description: Version of equsal 2291 with a dv condition, which does not require ax-13 2246. See equsalvw 1931 for a version with two dv conditions requiring fewer axioms. See also the dual form equsexv 2109. (Contributed by BJ, 31-May-2019.)

Hypotheses

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

Assertion

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

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem equsalv

Step	Hyp	Ref	Expression
1		equsalv.nf	. . 3 |- F/ x ps
2	1	19.23 2080	. 2 |- ( A. x ( x = y -> ps ) <-> ( E. x x = y -> ps ) )
3		equsalv.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
4	3	pm5.74i 260	. . 3 |- ( ( x = y -> ph ) <-> ( x = y -> ps ) )
5	4	albii 1747	. 2 |- ( A. x ( x = y -> ph ) <-> A. x ( x = y -> ps ) )
6		ax6ev 1890	. . 3 |- E. x x = y
7	6	a1bi 352	. 2 |- ( ps <-> ( E. x x = y -> ps ) )
8	2, 5, 7	3bitr4i 292	1 |- ( A. x ( x = y -> ph ) <-> ps )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704   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

This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1705  df-nf 1710

This theorem is referenced by:  bj-equsalhv  32744