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

Theorem exlimddv 1628
Description: Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
Hypotheses
Ref Expression
exlimddv.1  |-  ( ph  ->  E. x ps )
exlimddv.2  |-  ( (
ph  /\  ps )  ->  ch )
Assertion
Ref Expression
exlimddv  |-  ( ph  ->  ch )
Distinct variable groups:    ch, x    ph, x
Allowed substitution hint:    ps( x)

Proof of Theorem exlimddv
StepHypRef Expression
1 exlimddv.1 . 2  |-  ( ph  ->  E. x ps )
2 exlimddv.2 . . . 4  |-  ( (
ph  /\  ps )  ->  ch )
32ex 423 . . 3  |-  ( ph  ->  ( ps  ->  ch ) )
43exlimdv 1626 . 2  |-  ( ph  ->  ( E. x ps 
->  ch ) )
51, 4mpd 14 1  |-  ( ph  ->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1531
This theorem is referenced by:  mrieqv2d  13557  mreexexlem4d  13565  acsinfd  14299  acsdomd  14300
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
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
  Copyright terms: Public domain W3C validator