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  5812  poxp  6400  rdgss  6552  dom2lem  6948  supisoti  7212  ordiso2  7237  updjud  7284  uzind  9594  zindd  9601  lbzbi  9853  icoshftf1o  10229  ccatrn  11193  ccatalpha  11197  maxabslemval  11789  xrmaxiflemval  11831  fisum0diag2  12029  alzdvds  12436  hashgcdeq  12833  ghmrn  13865  ghmpreima  13874  imasring  14099  01eq0ring  14225  islssmd  14395  tgcl  14815  distop  14836  neiuni  14912  cnpnei  14970  isxmetd  15098  fsumcncntop  15318  fsumdvdsmul  15742  uspgr2wlkeq  16243  clwwlkccatlem  16278  bj-nntrans2  16606  bj-inf2vnlem1  16624