Theorem ralimdv2 2695

Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 1-Feb-2005.)

Hypothesis

Ref	Expression
ralimdv2.1	|- (ph -> ((x e. A -> ps) -> (x e. B -> ch)))

Assertion

Ref	Expression
ralimdv2	|- (ph -> (A.x e. A ps -> A.x e. B ch))

Distinct variable group:   ph,x

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

Proof of Theorem ralimdv2

Step	Hyp	Ref	Expression
1		ralimdv2.1	. . 3 |- (ph -> ((x e. A -> ps) -> (x e. B -> ch)))
2	1	alimdv 1621	. 2 |- (ph -> (A.x(x e. A -> ps) -> A.x(x e. B -> ch)))
3		df-ral 2620	. 2 |- (A.x e. A ps <-> A.x(x e. A -> ps))
4		df-ral 2620	. 2 |- (A.x e. B ch <-> A.x(x e. B -> ch))
5	2, 3, 4	3imtr4g 261	1 |- (ph -> (A.x e. A ps -> A.x e. B ch))

Syntax hints:   -> wi 4  A.wal 1540   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

This theorem depends on definitions:  df-bi 177  df-ral 2620

This theorem is referenced by:  ssralv  3331