Theorem bj-cbv2hv 32731

Description: Version of cbv2h 2269 with a dv condition, which does not require ax-13 2246. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-cbv2hv.1	|- ( ph -> ( ps -> A. y ps ) )
bj-cbv2hv.2	|- ( ph -> ( ch -> A. x ch ) )
bj-cbv2hv.3	|- ( ph -> ( x = y -> ( ps <-> ch ) ) )

Assertion

Ref	Expression
bj-cbv2hv	|- ( A. x A. y ph -> ( A. x ps <-> A. y ch ) )

Distinct variable group:   x, y

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

Proof of Theorem bj-cbv2hv

Step	Hyp	Ref	Expression
1		bj-cbv2hv.1	. . 3 |- ( ph -> ( ps -> A. y ps ) )
2		bj-cbv2hv.2	. . 3 |- ( ph -> ( ch -> A. x ch ) )
3		bj-cbv2hv.3	. . . 4 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
4		biimp 205	. . . 4 |- ( ( ps <-> ch ) -> ( ps -> ch ) )
5	3, 4	syl6 35	. . 3 |- ( ph -> ( x = y -> ( ps -> ch ) ) )
6	1, 2, 5	bj-cbv1hv 32730	. 2 |- ( A. x A. y ph -> ( A. x ps -> A. y ch ) )
7		equcomi 1944	. . . . 5 |- ( y = x -> x = y )
8		biimpr 210	. . . . 5 |- ( ( ps <-> ch ) -> ( ch -> ps ) )
9	7, 3, 8	syl56 36	. . . 4 |- ( ph -> ( y = x -> ( ch -> ps ) ) )
10	2, 1, 9	bj-cbv1hv 32730	. . 3 |- ( A. y A. x ph -> ( A. y ch -> A. x ps ) )
11	10	alcoms 2035	. 2 |- ( A. x A. y ph -> ( A. y ch -> A. x ps ) )
12	6, 11	impbid 202	1 |- ( A. x A. y ph -> ( A. x ps <-> A. y ch ) )

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

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-cbv2v  32732