Theorem rexlimd2 3025

Description: Version of rexlimd 3026 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 2957	. 2 |- ( ph -> A. x e. A ( ps -> ch ) )
4		rexlimd2.2	. . 3 |- ( ph -> F/ x ch )
5		r19.23t 3021	. . 3 |- ( F/ x ch -> ( A. x e. A ( ps -> ch ) <-> ( E. x e. A ps -> ch ) ) )
6	4, 5	syl 17	. 2 |- ( ph -> ( A. x e. A ( ps -> ch ) <-> ( E. x e. A ps -> ch ) ) )
7	3, 6	mpbid 222	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   <-> wb 196   F/wnf 1708   e. wcel 1990   A.wral 2912   E.wrex 2913

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-or 385  df-an 386  df-ex 1705  df-nf 1710  df-ral 2917  df-rex 2918

This theorem is referenced by:  rexlimd  3026  sbcrext  3511  sbcrextOLD  3512