Theorem bj-spimev 32720

Description: Version of spime 2256 with a dv condition, which does not require ax-13 2246. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-spimev.1	|- F/ x ph
bj-spimev.2	|- ( x = y -> ( ph -> ps ) )

Assertion

Ref	Expression
bj-spimev	|- ( ph -> E. x ps )

Distinct variable group:   x, y

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

Proof of Theorem bj-spimev

Step	Hyp	Ref	Expression
1		bj-spimev.1	. . . 4 |- F/ x ph
2	1	a1i 11	. . 3 |- ( T. -> F/ x ph )
3		bj-spimev.2	. . 3 |- ( x = y -> ( ph -> ps ) )
4	2, 3	bj-spimedv 32719	. 2 |- ( T. -> ( ph -> E. x ps ) )
5	4	trud 1493	1 |- ( ph -> E. x ps )

Syntax hints:   -> wi 4   T. wtru 1484   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-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by:  bj-spimevv  32722