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Theorem ralrimd 2631
Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 16-Feb-2004.)
Hypotheses
Ref Expression
ralrimd.1  |-  F/ x ph
ralrimd.2  |-  F/ x ps
ralrimd.3  |-  ( ph  ->  ( ps  ->  (
x  e.  A  ->  ch ) ) )
Assertion
Ref Expression
ralrimd  |-  ( ph  ->  ( ps  ->  A. x  e.  A  ch )
)

Proof of Theorem ralrimd
StepHypRef Expression
1 ralrimd.1 . . 3  |-  F/ x ph
2 ralrimd.2 . . 3  |-  F/ x ps
3 ralrimd.3 . . 3  |-  ( ph  ->  ( ps  ->  (
x  e.  A  ->  ch ) ) )
41, 2, 3alrimd 1749 . 2  |-  ( ph  ->  ( ps  ->  A. x
( x  e.  A  ->  ch ) ) )
5 df-ral 2548 . 2  |-  ( A. x  e.  A  ch  <->  A. x ( x  e.  A  ->  ch )
)
64, 5syl6ibr 218 1  |-  ( ph  ->  ( ps  ->  A. x  e.  A  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527   F/wnf 1531    e. wcel 1684   A.wral 2543
This theorem is referenced by:  ralrimdv  2632  reusv2lem3  4537  fliftfun  5811  riotasv3d  6353  mapxpen  7027  domtriomlem  8068  fzrevral  10866  dedekind  24082  ssralv2  28294
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-nf 1532  df-ral 2548
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