Theorem ralimdv 2431

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 8-Oct-2003.)

Hypothesis

Ref	Expression
ralimdv.1	|- ( ph -> ( ps -> ch ) )

Assertion

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

Distinct variable group:   ph, x

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

Proof of Theorem ralimdv

Step	Hyp	Ref	Expression
1		ralimdv.1	. . 3 |- ( ph -> ( ps -> ch ) )
2	1	adantr 270	. 2 |- ( ( ph /\ x e. A ) -> ( ps -> ch ) )
3	2	ralimdva 2430	1 |- ( ph -> ( A. x e. A ps -> A. x e. A ch ) )

Syntax hints:   -> wi 4   e. wcel 1434   A.wral 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-4 1441  ax-17 1460

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-ral 2354

This theorem is referenced by:  poss  4061  sess1  4100  sess2  4101  riinint  4621  dffo4  5347  dffo5  5348  isoini2  5489  rdgivallem  6030  iinerm  6244  xpf1o  6385  resqrexlemgt0  10044  cau3lem  10138  caubnd2  10141  climshftlemg  10279  climcau  10322  climcaucn  10326  serif0  10327  bezoutlemmain  10531