Theorem bj-drex1v 32749

Description: Version of drex1 2327 with a dv condition, which does not require ax-13 2246. (Contributed by BJ, 17-Jun-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-drex1v.1	|- ( A. x x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
bj-drex1v	|- ( A. x x = y -> ( E. x ph <-> E. y ps ) )

Distinct variable group:   x, y

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

Proof of Theorem bj-drex1v

Step	Hyp	Ref	Expression
1		bj-drex1v.1	. . . . 5 |- ( A. x x = y -> ( ph <-> ps ) )
2	1	notbid 308	. . . 4 |- ( A. x x = y -> ( -. ph <-> -. ps ) )
3	2	bj-dral1v 32748	. . 3 |- ( A. x x = y -> ( A. x -. ph <-> A. y -. ps ) )
4	3	notbid 308	. 2 |- ( A. x x = y -> ( -. A. x -. ph <-> -. A. y -. ps ) )
5		df-ex 1705	. 2 |- ( E. x ph <-> -. A. x -. ph )
6		df-ex 1705	. 2 |- ( E. y ps <-> -. A. y -. ps )
7	4, 5, 6	3bitr4g 303	1 |- ( A. x x = y -> ( E. x ph <-> E. y ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   A.wal 1481   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-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)