Theorem ralimi2 2687

Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 22-Feb-2004.)

Hypothesis

Ref	Expression
ralimi2.1	|- ((x e. A -> ph) -> (x e. B -> ps))

Assertion

Ref	Expression
ralimi2	|- (A.x e. A ph -> A.x e. B ps)

Proof of Theorem ralimi2

Step	Hyp	Ref	Expression
1		ralimi2.1	. . 3 |- ((x e. A -> ph) -> (x e. B -> ps))
2	1	alimi 1559	. 2 |- (A.x(x e. A -> ph) -> A.x(x e. B -> ps))
3		df-ral 2620	. 2 |- (A.x e. A ph <-> A.x(x e. A -> ph))
4		df-ral 2620	. 2 |- (A.x e. B ps <-> A.x(x e. B -> ps))
5	2, 3, 4	3imtr4i 257	1 |- (A.x e. A ph -> A.x e. B ps)

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

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

This theorem is referenced by:  ralimia  2688  ralcom3  2777  peano5  4410