Theorem ralimda 39326

Description: Deduction quantifying both antecedent and consequent. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
ralimda.1	|- F/ x ph
ralimda.2	|- ( ( ph /\ x e. A ) -> ( ps -> ch ) )

Assertion

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

Proof of Theorem ralimda

Step	Hyp	Ref	Expression
1		ralimda.1	. . . 4 |- F/ x ph
2		nfra1 2941	. . . 4 |- F/ x A. x e. A ps
3	1, 2	nfan 1828	. . 3 |- F/ x ( ph /\ A. x e. A ps )
4		id 22	. . . . 5 |- ( ( ph /\ x e. A ) -> ( ph /\ x e. A ) )
5	4	adantlr 751	. . . 4 |- ( ( ( ph /\ A. x e. A ps ) /\ x e. A ) -> ( ph /\ x e. A ) )
6		rspa 2930	. . . . 5 |- ( ( A. x e. A ps /\ x e. A ) -> ps )
7	6	adantll 750	. . . 4 |- ( ( ( ph /\ A. x e. A ps ) /\ x e. A ) -> ps )
8		ralimda.2	. . . 4 |- ( ( ph /\ x e. A ) -> ( ps -> ch ) )
9	5, 7, 8	sylc 65	. . 3 |- ( ( ( ph /\ A. x e. A ps ) /\ x e. A ) -> ch )
10	3, 9	ralrimia 39315	. 2 |- ( ( ph /\ A. x e. A ps ) -> A. x e. A ch )
11	10	ex 450	1 |- ( ph -> ( A. x e. A ps -> A. x e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 384   F/wnf 1708   e. wcel 1990   A.wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-ral 2917

This theorem is referenced by:  xlimmnfvlem1  40058  xlimmnfvlem2  40059  xlimpnfvlem1  40062  xlimpnfvlem2  40063