Theorem rexlimd 2736

Description: Deduction from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Hypotheses

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

Assertion

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

Proof of Theorem rexlimd

Step	Hyp	Ref	Expression
1		rexlimd.1	. . 3 |- F/xph
2		rexlimd.3	. . 3 |- (ph -> (x e. A -> (ps -> ch)))
3	1, 2	ralrimi 2696	. 2 |- (ph -> A.x e. A (ps -> ch))
4		rexlimd.2	. . 3 |- F/xch
5	4	r19.23 2730	. 2 |- (A.x e. A (ps -> ch) <-> (E.x e. A ps -> ch))
6	3, 5	sylib 188	1 |- (ph -> (E.x e. A ps -> ch))

Syntax hints:   -> wi 4   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:  rexlimdv  2738  fun11iun  5306  ffnfv  5428