Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ceqsexv2d Unicode version

Theorem ceqsexv2d 23178
Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016.)
Hypotheses
Ref Expression
ceqsexv2d.1  |-  A  e. 
_V
ceqsexv2d.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
ceqsexv2d.3  |-  ps
Assertion
Ref Expression
ceqsexv2d  |-  E. x ph
Distinct variable groups:    x, A    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem ceqsexv2d
StepHypRef Expression
1 ceqsexv2d.3 . 2  |-  ps
2 ceqsexv2d.1 . . . 4  |-  A  e. 
_V
3 ceqsexv2d.2 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
42, 3ceqsexv 2836 . . 3  |-  ( E. x ( x  =  A  /\  ph )  <->  ps )
54biimpri 197 . 2  |-  ( ps 
->  E. x ( x  =  A  /\  ph ) )
6 simpr 447 . . 3  |-  ( ( x  =  A  /\  ph )  ->  ph )
76eximi 1566 . 2  |-  ( E. x ( x  =  A  /\  ph )  ->  E. x ph )
81, 5, 7mp2b 9 1  |-  E. x ph
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801
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-6 1715  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-v 2803
  Copyright terms: Public domain W3C validator