Theorem ralrimdva 2399

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Feb-2008.)

Hypothesis

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

Assertion

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

Distinct variable groups:   ph, x   ps, x

Allowed substitution hints:   ch( x)   A( x)

Proof of Theorem ralrimdva

Step	Hyp	Ref	Expression
1		ralrimdva.1	. . . 4 |- ( ( ph /\ x e. A ) -> ( ps -> ch ) )
2	1	ex 108	. . 3 |- ( ph -> ( x e. A -> ( ps -> ch ) ) )
3	2	com23 72	. 2 |- ( ph -> ( ps -> ( x e. A -> ch ) ) )
4	3	ralrimdv 2398	1 |- ( ph -> ( ps -> A. x e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 97   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  ax-4 1400  ax-17 1419

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311

This theorem is referenced by:  ralxfrd  4194  isoselem  5459  isosolem  5463  findcard  6345  nnsub  7952  ublbneg  8548  expnlbnd2  9374  cau3lem  9710  climshftlemg  9823  subcn2  9832  serif0  9871  sqrt2irr  9878