Theorem ralimi2 2381

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 1344	. 2 |- ( A. x ( x e. A -> ph ) -> A. x ( x e. B -> ps ) )
3		df-ral 2311	. 2 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
4		df-ral 2311	. 2 |- ( A. x e. B ps <-> A. x ( x e. B -> ps ) )
5	2, 3, 4	3imtr4i 190	1 |- ( A. x e. A ph -> A. x e. B ps )

Syntax hints:   -> wi 4   A.wal 1241   e. wcel 1393   A.wral 2306

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338

This theorem depends on definitions:  df-bi 110  df-ral 2311

This theorem is referenced by:  ralimia  2382  ralcom3  2477  bj-nntrans  10076  bj-findis  10104