MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ralimdaa Structured version   Unicode version

Theorem ralimdaa 2826
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, 4-Dec-2019.) Shortened after introduction of hbralimdaa 2825. (Revised by Wolf Lammen, 5-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
21nfri 1813 . 2  |-  ( ph  ->  A. x ph )
3 ralimdaa.2 . 2  |-  ( (
ph  /\  x  e.  A )  ->  ( ps  ->  ch ) )
42, 3hbralimdaa 2825 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 1590    e. wcel 1758   A.wral 2799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-12 1794
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591  df-ral 2804
This theorem is referenced by:  ralimdvaOLD  2833  eltsk2g  9030  ptcnplem  19327  infrglb  29920  stoweidlem61  30005  stoweid  30007
  Copyright terms: Public domain W3C validator