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Theorem ralimdaa 2806
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 432 . . 3  |-  ( ph  ->  ( x  e.  A  ->  ( ps  ->  ch ) ) )
41, 3ralrimi 2804 . 2  |-  ( ph  ->  A. x  e.  A  ( ps  ->  ch )
)
5 ralim 2793 . 2  |-  ( A. x  e.  A  ( ps  ->  ch )  -> 
( A. x  e.  A  ps  ->  A. x  e.  A  ch )
)
64, 5syl 17 1  |-  ( ph  ->  ( A. x  e.  A  ps  ->  A. x  e.  A  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   F/wnf 1637    e. wcel 1842   A.wral 2754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-12 1878
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1634  df-nf 1638  df-ral 2759
This theorem is referenced by:  ralimdvaOLD  2813  eltsk2g  9159  ptcnplem  20414  infrglb  36952  stoweidlem61  37211  stoweid  37213  fourierdlem73  37330
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