Theorem equsexv 2109

Description: Version of equsex 2292 with a dv condition, which does not require ax-13 2246. See equsexvw 1932 for a version with two dv conditions requiring fewer axioms. See also the dual form equsalv 2108. (Contributed by BJ, 31-May-2019.)

Hypotheses

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

Assertion

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

Distinct variable group:   x, y

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

Proof of Theorem equsexv

Step	Hyp	Ref	Expression
1		equsalv.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		equsalv.nf	. . . 4 |- F/ x ps
6	5	19.41 2103	. . 3 |- ( E. x ( x = y /\ ps ) <-> ( E. x x = y /\ ps ) )
7	4, 6	mpbiran 953	. 2 |- ( E. x ( x = y /\ ps ) <-> ps )
8	3, 7	bitri 264	1 |- ( E. x ( x = y /\ ph ) <-> ps )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   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-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  equsexhv  2124  sb56  2150  cleljustALT2  2186  sb10f  2456  dprd2d2  18443  poimirlem25  33434