Theorem rexlimd2 2737

Description: Version of rexlimd 2736 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/xph
rexlimd2.2	|- (ph -> F/xch)
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/xph
2		rexlimd2.3	. . 3 |- (ph -> (x e. A -> (ps -> ch)))
3	1, 2	ralrimi 2696	. 2 |- (ph -> A.x e. A (ps -> ch))
4		rexlimd2.2	. . 3 |- (ph -> F/xch)
5		r19.23t 2729	. . 3 |- ( F/xch -> (A.x e. A (ps -> ch) <-> (E.x e. A ps -> ch)))
6	4, 5	syl 15	. 2 |- (ph -> (A.x e. A (ps -> ch) <-> (E.x e. A ps -> ch)))
7	3, 6	mpbid 201	1 |- (ph -> (E.x e. A ps -> ch))

Syntax hints:   -> wi 4   <-> wb 176   F/wnf 1544   e. wcel 1710  A.wral 2615  E.wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2620  df-rex 2621

This theorem is referenced by: (None)