Theorem rexlimdv3a 2585

Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 2582. (Contributed by NM, 7-Jun-2015.)

Hypothesis

Ref	Expression
rexlimdv3a.1	|- ( ( ph /\ x e. A /\ ps ) -> ch )

Assertion

Ref	Expression
rexlimdv3a	|- ( ph -> ( E. x e. A ps -> ch ) )

Distinct variable groups:   ph, x   ch, x

Allowed substitution hints:   ps( x)   A( x)

Proof of Theorem rexlimdv3a

Step	Hyp	Ref	Expression
1		rexlimdv3a.1	. . 3 |- ( ( ph /\ x e. A /\ ps ) -> ch )
2	1	3exp 1192	. 2 |- ( ph -> ( x e. A -> ( ps -> ch ) ) )
3	2	rexlimdv 2582	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   /\ w3a 968   e. wcel 2136   E.wrex 2445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-3an 970  df-nf 1449  df-ral 2449  df-rex 2450

This theorem is referenced by:  resqrtcl  10971