MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reximd2a Structured version   Unicode version

Theorem reximd2a 2913
Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 27-Jan-2020.)
Hypotheses
Ref Expression
reximd2a.1  |-  F/ x ph
reximd2a.2  |-  ( ( ( ph  /\  x  e.  A )  /\  ps )  ->  x  e.  B
)
reximd2a.3  |-  ( ( ( ph  /\  x  e.  A )  /\  ps )  ->  ch )
reximd2a.4  |-  ( ph  ->  E. x  e.  A  ps )
Assertion
Ref Expression
reximd2a  |-  ( ph  ->  E. x  e.  B  ch )

Proof of Theorem reximd2a
StepHypRef Expression
1 reximd2a.4 . 2  |-  ( ph  ->  E. x  e.  A  ps )
2 reximd2a.1 . . . 4  |-  F/ x ph
3 reximd2a.2 . . . . . 6  |-  ( ( ( ph  /\  x  e.  A )  /\  ps )  ->  x  e.  B
)
4 reximd2a.3 . . . . . 6  |-  ( ( ( ph  /\  x  e.  A )  /\  ps )  ->  ch )
53, 4jca 532 . . . . 5  |-  ( ( ( ph  /\  x  e.  A )  /\  ps )  ->  ( x  e.  B  /\  ch )
)
65expl 618 . . . 4  |-  ( ph  ->  ( ( x  e.  A  /\  ps )  ->  ( x  e.  B  /\  ch ) ) )
72, 6eximd 1868 . . 3  |-  ( ph  ->  ( E. x ( x  e.  A  /\  ps )  ->  E. x
( x  e.  B  /\  ch ) ) )
8 df-rex 2799 . . 3  |-  ( E. x  e.  A  ps  <->  E. x ( x  e.  A  /\  ps )
)
9 df-rex 2799 . . 3  |-  ( E. x  e.  B  ch  <->  E. x ( x  e.  B  /\  ch )
)
107, 8, 93imtr4g 270 . 2  |-  ( ph  ->  ( E. x  e.  A  ps  ->  E. x  e.  B  ch )
)
111, 10mpd 15 1  |-  ( ph  ->  E. x  e.  B  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   E.wex 1599   F/wnf 1603    e. wcel 1804   E.wrex 2794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-12 1840
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1600  df-nf 1604  df-rex 2799
This theorem is referenced by:  locfinreflem  27821
  Copyright terms: Public domain W3C validator