Theorem r19.20i2 1252

Description: Inference quantifying both antecedent and consequent.

Hypothesis

Ref	Expression
r19.20i2.1	|- ((x e. A -> ph) -> (x e. B -> ps))

Assertion

Ref	Expression
r19.20i2	|- (A.x e. A ph -> A.x e. B ps)

Proof of Theorem r19.20i2

Step	Hyp	Ref	Expression
1		r19.20i2.1	. . 3 |- ((x e. A -> ph) -> (x e. B -> ps))
2	1	19.20i 691	. 2 |- (A.x(x e. A -> ph) -> A.x(x e. B -> ps))
3		df-ral 1205	. 2 |- (A.x e. A ph <-> A.x(x e. A -> ph))
4		df-ral 1205	. 2 |- (A.x e. B ps <-> A.x(x e. B -> ps))
5	2, 3, 4	3imtr4 192	1 |- (A.x e. A ph -> A.x e. B ps)

Syntax hints:   -> wi 2  A.wal 672   e. wcel 1092  A.wral 1201

This theorem is referenced by:  r19.20i 1253  ralcom3 1315  omex 3475  kmlem1 3580

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-ral 1205

metamath.org