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Theorem ralrimd 2836
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 1934 . 2  |-  ( ph  ->  ( ps  ->  A. x
( x  e.  A  ->  ch ) ) )
5 df-ral 2787 . 2  |-  ( A. x  e.  A  ch  <->  A. x ( x  e.  A  ->  ch )
)
64, 5syl6ibr 230 1  |-  ( ph  ->  ( ps  ->  A. x  e.  A  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1435   F/wnf 1663    e. wcel 1870   A.wral 2782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-12 1907
This theorem depends on definitions:  df-bi 188  df-ex 1660  df-nf 1664  df-ral 2787
This theorem is referenced by:  ralrimdvOLD  2849  reusv2lem3  4628  fliftfun  6220  mapxpen  7744  domtriomlem  8870  dedekind  9796  fzrevral  11877  riotasv3d  32241  ssralv2  36525
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