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Theorem ralimdaa 2633
Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-Sep-2003.)
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 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  ( ps  ->  ch ) )
32ex 423 . . . 4  |-  ( ph  ->  ( x  e.  A  ->  ( ps  ->  ch ) ) )
43a2d 23 . . 3  |-  ( ph  ->  ( ( x  e.  A  ->  ps )  ->  ( x  e.  A  ->  ch ) ) )
51, 4alimd 1756 . 2  |-  ( ph  ->  ( A. x ( x  e.  A  ->  ps )  ->  A. x
( x  e.  A  ->  ch ) ) )
6 df-ral 2561 . 2  |-  ( A. x  e.  A  ps  <->  A. x ( x  e.  A  ->  ps )
)
7 df-ral 2561 . 2  |-  ( A. x  e.  A  ch  <->  A. x ( x  e.  A  ->  ch )
)
85, 6, 73imtr4g 261 1  |-  ( ph  ->  ( A. x  e.  A  ps  ->  A. x  e.  A  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1530   F/wnf 1534    e. wcel 1696   A.wral 2556
This theorem is referenced by:  ralimdva  2634  eltsk2g  8389  ptcnplem  17331  infrglb  27825
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-an 360  df-nf 1535  df-ral 2561
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