Theorem bj-cbvexdvav 32742

Description: Version of cbvexdva 2283 with a dv condition, which does not require ax-13 2246. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-cbvaldvav.1	|- ( ( ph /\ x = y ) -> ( ps <-> ch ) )

Assertion

Ref	Expression
bj-cbvexdvav	|- ( ph -> ( E. x ps <-> E. y ch ) )

Distinct variable groups:   ps, y   ch, x   ph, x, y

Allowed substitution hints:   ps( x)   ch( y)

Proof of Theorem bj-cbvexdvav

Step	Hyp	Ref	Expression
1		nfv 1843	. 2 |- F/ y ph
2		nfvd 1844	. 2 |- ( ph -> F/ y ps )
3		bj-cbvaldvav.1	. . 3 |- ( ( ph /\ x = y ) -> ( ps <-> ch ) )
4	3	ex 450	. 2 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
5	1, 2, 4	bj-cbvexdv 32736	1 |- ( ph -> ( E. x ps <-> E. y ch ) )

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  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)