Theorem equsexvw 1932

Description: Version of equsexv 2109 with a dv condition, and of equsex 2292 with two dv conditions, which requires fewer axioms. See also the dual form equsalvw 1931. (Contributed by BJ, 31-May-2019.)

Hypothesis

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

Assertion

Ref	Expression
equsexvw	|- ( E. x ( x = y /\ ph ) <-> ps )

Distinct variable groups:   x, y   ps, x

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

Proof of Theorem equsexvw

Step	Hyp	Ref	Expression
1		equsalvw.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
2	1	pm5.32i 669	. . 3 |- ( ( x = y /\ ph ) <-> ( x = y /\ ps ) )
3	2	exbii 1774	. 2 |- ( E. x ( x = y /\ ph ) <-> E. x ( x = y /\ ps ) )
4		ax6ev 1890	. . 3 |- E. x x = y
5		19.41v 1914	. . 3 |- ( E. x ( x = y /\ ps ) <-> ( E. x x = y /\ ps ) )
6	4, 5	mpbiran 953	. 2 |- ( E. x ( x = y /\ ps ) <-> ps )
7	3, 6	bitri 264	1 |- ( E. x ( x = y /\ ph ) <-> ps )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   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-an 386  df-ex 1705

This theorem is referenced by:  cleljust  1998  sbhypf  3253  axsep  4780  dfid3  5025  opeliunxp  5170  imai  5478  coi1  5651  elfuns  32022  bj-axsep  32793