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Theorem ralrimd 2644
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 1761 . 2  |-  ( ph  ->  ( ps  ->  A. x
( x  e.  A  ->  ch ) ) )
5 df-ral 2561 . 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 1530   F/wnf 1534    e. wcel 1696   A.wral 2556
This theorem is referenced by:  ralrimdv  2645  reusv2lem3  4553  fliftfun  5827  riotasv3d  6369  mapxpen  7043  domtriomlem  8084  fzrevral  10882  dedekind  24097  ssralv2  28593
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-nf 1535  df-ral 2561
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