Theorem ralrimiv 2604

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.)

Hypothesis

Ref	Expression
ralrimiv.1	|- ( ph -> ( x e. A -> ps ) )

Assertion

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

Distinct variable group:   ph, x

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

Proof of Theorem ralrimiv

Step	Hyp	Ref	Expression
1		nfv 1576	. 2 |- F/ x ph
2		ralrimiv.1	. 2 |- ( ph -> ( x e. A -> ps ) )
3	1, 2	ralrimi 2603	1 |- ( ph -> A. x e. A ps )

Syntax hints:   -> wi 4   e. wcel 2202   A.wral 2510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1495  ax-gen 1497  ax-4 1558  ax-17 1574

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-ral 2515

This theorem is referenced by:  ralrimiva  2605  ralrimivw  2606  ralrimivv  2613  r19.27av  2668  rr19.3v  2945  rabssdv  3307  rzal  3592  trin  4197  class2seteq  4253  ralxfrALT  4564  ssorduni  4585  ordsucim  4598  onintonm  4615  issref  5119  funimaexglem  5413  resflem  5811  poxp  6397  rdgss  6549  dom2lem  6945  supisoti  7209  ordiso2  7234  updjud  7281  uzind  9591  zindd  9598  lbzbi  9850  icoshftf1o  10226  ccatrn  11186  ccatalpha  11190  maxabslemval  11769  xrmaxiflemval  11811  fisum0diag2  12009  alzdvds  12416  hashgcdeq  12813  ghmrn  13845  ghmpreima  13854  imasring  14079  01eq0ring  14205  islssmd  14375  tgcl  14790  distop  14811  neiuni  14887  cnpnei  14945  isxmetd  15073  fsumcncntop  15293  fsumdvdsmul  15717  uspgr2wlkeq  16218  clwwlkccatlem  16253  bj-nntrans2  16550  bj-inf2vnlem1  16568