Theorem equsalvw 1931

Description: Version of equsalv 2108 with a dv condition, and of equsal 2291 with two dv conditions, which requires fewer axioms. See also the dual form equsexvw 1932. (Contributed by BJ, 31-May-2019.)

Hypothesis

Ref	Expression
equsalvw.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

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

Distinct variable groups:   x, y   ps, x

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

Proof of Theorem equsalvw

Step	Hyp	Ref	Expression
1		19.23v 1902	. 2 |- ( A. x ( x = y -> ps ) <-> ( E. x x = y -> ps ) )
2		equsalvw.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
3	2	pm5.74i 260	. . 3 |- ( ( x = y -> ph ) <-> ( x = y -> ps ) )
4	3	albii 1747	. 2 |- ( A. x ( x = y -> ph ) <-> A. x ( x = y -> ps ) )
5		ax6ev 1890	. . 3 |- E. x x = y
6	5	a1bi 352	. 2 |- ( ps <-> ( E. x x = y -> ps ) )
7	1, 4, 6	3bitr4i 292	1 |- ( A. x ( x = y -> ph ) <-> ps )

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

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

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  ax13lem2  2296  reu8  3402  asymref2  5513  intirr  5514  fun11  5963  bj-dvelimdv  32834  bj-dvelimdv1  32835  wl-clelv2-just  33379  undmrnresiss  37910  pm13.192  38611