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Theorem ralimdaa 2866
Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-Sep-2003.) (Proof shortened by Wolf Lammen, 29-Dec-2019.)
Hypotheses
Ref Expression
ralimdaa.1  |-  F/ x ph
ralimdaa.2  |-  ( (
ph  /\  x  e.  A )  ->  ( ps  ->  ch ) )
Assertion
Ref Expression
ralimdaa  |-  ( ph  ->  ( A. x  e.  A  ps  ->  A. x  e.  A  ch )
)

Proof of Theorem ralimdaa
StepHypRef Expression
1 ralimdaa.1 . . 3  |-  F/ x ph
2 ralimdaa.2 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( ps  ->  ch ) )
32ex 434 . . 3  |-  ( ph  ->  ( x  e.  A  ->  ( ps  ->  ch ) ) )
41, 3ralrimi 2864 . 2  |-  ( ph  ->  A. x  e.  A  ( ps  ->  ch )
)
5 ralim 2853 . 2  |-  ( A. x  e.  A  ( ps  ->  ch )  -> 
( A. x  e.  A  ps  ->  A. x  e.  A  ch )
)
64, 5syl 16 1  |-  ( ph  ->  ( A. x  e.  A  ps  ->  A. x  e.  A  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   F/wnf 1599    e. wcel 1767   A.wral 2814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-12 1803
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600  df-ral 2819
This theorem is referenced by:  ralimdvaOLD  2873  eltsk2g  9125  ptcnplem  19854  infrglb  31140  stoweidlem61  31361  stoweid  31363  fourierdlem73  31480
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