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Theorem ralrimiv 2481
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
StepHypRef Expression
1 nfv 1493 . 2  |-  F/ x ph
2 ralrimiv.1 . 2  |-  ( ph  ->  ( x  e.  A  ->  ps ) )
31, 2ralrimi 2480 1  |-  ( ph  ->  A. x  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1465   A.wral 2393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-4 1472  ax-17 1491
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-ral 2398
This theorem is referenced by:  ralrimiva  2482  ralrimivw  2483  ralrimivv  2490  r19.27av  2544  rr19.3v  2797  rabssdv  3147  rzal  3430  trin  4006  class2seteq  4057  ralxfrALT  4358  ssorduni  4373  ordsucim  4386  onintonm  4403  issref  4891  funimaexglem  5176  resflem  5552  poxp  6097  rdgss  6248  dom2lem  6634  supisoti  6865  ordiso2  6888  updjud  6935  uzind  9130  zindd  9137  lbzbi  9376  icoshftf1o  9742  maxabslemval  10948  xrmaxiflemval  10987  fisum0diag2  11184  alzdvds  11479  hashgcdeq  11831  tgcl  12160  distop  12181  neiuni  12257  cnpnei  12315  isxmetd  12443  fsumcncntop  12652  bj-nntrans2  13077  bj-inf2vnlem1  13095
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