Theorem bj-cbvexvv 32734

Description: Version of cbvexv 2275 with a dv condition, which does not require ax-13 2246. UPDATE: this is cbvexvw 1970 (which is proved with fewer axioms). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)

Hypothesis

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

Assertion

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

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

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

Proof of Theorem bj-cbvexvv

Step	Hyp	Ref	Expression
1		nfv 1843	. 2 |- F/ y ph
2		nfv 1843	. 2 |- F/ x ps
3		bj-cbvalvv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvexv1 2176	1 |- ( E. x ph <-> E. y ps )

Syntax hints:   -> wi 4   <-> wb 196   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)