Theorem rexlimd2 2585

Description: Version of rexlimd 2584 with deduction version of second hypothesis. (Contributed by NM, 21-Jul-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)

Hypotheses

Ref	Expression
rexlimd2.1	|- F/ x ph
rexlimd2.2	|- ( ph -> F/ x ch )
rexlimd2.3	|- ( ph -> ( x e. A -> ( ps -> ch ) ) )

Assertion

Ref	Expression
rexlimd2	|- ( ph -> ( E. x e. A ps -> ch ) )

Proof of Theorem rexlimd2

Step	Hyp	Ref	Expression
1		rexlimd2.1	. . 3 |- F/ x ph
2		rexlimd2.3	. . 3 |- ( ph -> ( x e. A -> ( ps -> ch ) ) )
3	1, 2	ralrimi 2541	. 2 |- ( ph -> A. x e. A ( ps -> ch ) )
4		rexlimd2.2	. . 3 |- ( ph -> F/ x ch )
5		r19.23t 2577	. . 3 |- ( F/ x ch -> ( A. x e. A ( ps -> ch ) <-> ( E. x e. A ps -> ch ) ) )
6	4, 5	syl 14	. 2 |- ( ph -> ( A. x e. A ( ps -> ch ) <-> ( E. x e. A ps -> ch ) ) )
7	3, 6	mpbid 146	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   <-> wb 104   F/wnf 1453   e. wcel 2141   A.wral 2448   E.wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-ral 2453  df-rex 2454

This theorem is referenced by:  sbcrext  3032