Theorem ralimdva 2693

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.)

Hypothesis

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

Assertion

Ref	Expression
ralimdva	|- (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 ralimdva

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 |- F/xph
2		ralimdva.1	. 2 |- ((ph /\ x e. A) -> (ps -> ch))
3	1, 2	ralimdaa 2692	1 |- (ph -> (A.x e. A ps -> A.x e. A ch))

Syntax hints:   -> wi 4   /\ wa 358   e. wcel 1710  A.wral 2615

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

This theorem is referenced by:  ralimdv  2694  weds  5939  nclenn  6250  spacind  6288