Theorem rexlimi 2731

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 30-Nov-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Hypotheses

Ref	Expression
rexlimi.1	|- F/xps
rexlimi.2	|- (x e. A -> (ph -> ps))

Assertion

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

Proof of Theorem rexlimi

Step	Hyp	Ref	Expression
1		rexlimi.2	. . 3 |- (x e. A -> (ph -> ps))
2	1	rgen 2679	. 2 |- A.x e. A (ph -> ps)
3		rexlimi.1	. . 3 |- F/xps
4	3	r19.23 2729	. 2 |- (A.x e. A (ph -> ps) <-> (E.x e. A ph -> ps))
5	2, 4	mpbi 199	1 |- (E.x e. A ph -> ps)

Syntax hints:   -> wi 4   F/wnf 1544   e. wcel 1710  A.wral 2614  E.wrex 2615

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2619  df-rex 2620

This theorem is referenced by:  rexlimiv  2732  fun11iun  5305