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Theorem ralrimd 2809
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 1815 . 2  |-  ( ph  ->  ( ps  ->  A. x
( x  e.  A  ->  ch ) ) )
5 df-ral 2725 . 2  |-  ( A. x  e.  A  ch  <->  A. x ( x  e.  A  ->  ch )
)
64, 5syl6ibr 227 1  |-  ( ph  ->  ( ps  ->  A. x  e.  A  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1367   F/wnf 1589    e. wcel 1756   A.wral 2720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-12 1792
This theorem depends on definitions:  df-bi 185  df-ex 1587  df-nf 1590  df-ral 2725
This theorem is referenced by:  ralrimdv  2810  reusv2lem3  4500  fliftfun  6010  mapxpen  7482  domtriomlem  8616  dedekind  9538  fzrevral  11549  ssralv2  31241  riotasv3d  32616
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