Theorem reximdai 2575

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/ x ph
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/ x ph
2		reximdai.2	. . 3 |- ( ph -> ( x e. A -> ( ps -> ch ) ) )
3	1, 2	ralrimi 2548	. 2 |- ( ph -> A. x e. A ( ps -> ch ) )
4		rexim 2571	. 2 |- ( A. x e. A ( ps -> ch ) -> ( E. x e. A ps -> E. x e. A ch ) )
5	3, 4	syl 14	1 |- ( ph -> ( E. x e. A ps -> E. x e. A ch ) )

Syntax hints:   -> wi 4   F/wnf 1460   e. wcel 2148   A.wral 2455   E.wrex 2456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-ral 2460  df-rex 2461

This theorem is referenced by:  reximdvai  2577  bezoutlemstep  12000  isomninnlem  14817  ismkvnnlem  14839