Theorem 2ralimi 2614

Description: Inference quantifying both antecedent and consequent two times, with strong hypothesis. (Contributed by AV, 3-Dec-2021.)

Hypothesis

Ref	Expression
ralimi.1	|- ( ph -> ps )

Assertion

Ref	Expression
2ralimi	|- ( A. x e. A A. y e. B ph -> A. x e. A A. y e. B ps )

Proof of Theorem 2ralimi

Step	Hyp	Ref	Expression
1		ralimi.1	. . 3 |- ( ph -> ps )
2	1	ralimi 2613	. 2 |- ( A. y e. B ph -> A. y e. B ps )
3	2	ralimi 2613	1 |- ( A. x e. A A. y e. B ph -> A. x e. A A. y e. B ps )

This proof depends on syntax axioms:   -> wi 4   A.wral 2528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502

This proof depends on definitions:  df-bi 117  df-ral 2533

This theorem is used by:  dfgrp3me  13907  rmodislmodlem  14689  rmodislmod  14690  xmeteq0  15462  xmettri2  15464