Theorem cbvexv1 2176

Description: Version of cbvex 2272 with a dv condition, which does not require ax-13 2246. See cbvexvw 1970 for a version with two dv conditions, requiring fewer axioms, and cbvexv 2275 for another variant. (Contributed by BJ, 31-May-2019.)

Hypotheses

Ref	Expression
cbvalv1.nf1	|- F/ y ph
cbvalv1.nf2	|- F/ x ps
cbvalv1.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvexv1	|- ( E. x ph <-> E. y ps )

Distinct variable group:   x, y

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

Proof of Theorem cbvexv1

Step	Hyp	Ref	Expression
1		cbvalv1.nf1	. . . . 5 |- F/ y ph
2	1	nfn 1784	. . . 4 |- F/ y -. ph
3		cbvalv1.nf2	. . . . 5 |- F/ x ps
4	3	nfn 1784	. . . 4 |- F/ x -. ps
5		cbvalv1.1	. . . . 5 |- ( x = y -> ( ph <-> ps ) )
6	5	notbid 308	. . . 4 |- ( x = y -> ( -. ph <-> -. ps ) )
7	2, 4, 6	cbvalv1 2175	. . 3 |- ( A. x -. ph <-> A. y -. ps )
8	7	notbii 310	. 2 |- ( -. A. x -. ph <-> -. A. y -. ps )
9		df-ex 1705	. 2 |- ( E. x ph <-> -. A. x -. ph )
10		df-ex 1705	. 2 |- ( E. y ps <-> -. A. y -. ps )
11	8, 9, 10	3bitr4i 292	1 |- ( E. x ph <-> E. y ps )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   A.wal 1481   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-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:  bj-cbvexvv  32734  bj-axrep1  32788  bj-axrep2  32789  bj-axrep4  32791