Theorem reximdai 2723

Description: Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 31-Aug-1999.)

Hypotheses

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

Assertion

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

Proof of Theorem reximdai

Step	Hyp	Ref	Expression
1		reximdai.1	. . 3 |- F/xph
2		reximdai.2	. . 3 |- (ph -> (x e. A -> (ps -> ch)))
3	1, 2	ralrimi 2696	. 2 |- (ph -> A.x e. A (ps -> ch))
4		rexim 2719	. 2 |- (A.x e. A (ps -> ch) -> (E.x e. A ps -> E.x e. A ch))
5	3, 4	syl 15	1 |- (ph -> (E.x e. A ps -> E.x e. A 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-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:  reximdvai  2725  chfnrn  5400