Theorem ralimdaa 2958

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-Sep-2003.) (Proof shortened by Wolf Lammen, 29-Dec-2019.)

Hypotheses

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

Assertion

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

Proof of Theorem ralimdaa

Step	Hyp	Ref	Expression
1		ralimdaa.1	. . 3 |- F/ x ph
2		ralimdaa.2	. . . 4 |- ( ( ph /\ x e. A ) -> ( ps -> ch ) )
3	2	ex 450	. . 3 |- ( ph -> ( x e. A -> ( ps -> ch ) ) )
4	1, 3	ralrimi 2957	. 2 |- ( ph -> A. x e. A ( ps -> ch ) )
5		ralim 2948	. 2 |- ( A. x e. A ( ps -> ch ) -> ( A. x e. A ps -> A. x e. A ch ) )
6	4, 5	syl 17	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-12 2047

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

This theorem is referenced by:  eltsk2g  9573  ptcnplem  21424  poimirlem26  33435  allbutfifvre  39907  climleltrp  39908  fnlimabslt  39911  stoweidlem61  40278  stoweid  40280  fourierdlem73  40396  smflimlem2  40980