MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ceqsex2v Structured version   Unicode version

Theorem ceqsex2v 2993
Description: Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)
Hypotheses
Ref Expression
ceqsex2v.1  |-  A  e. 
_V
ceqsex2v.2  |-  B  e. 
_V
ceqsex2v.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
ceqsex2v.4  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
ceqsex2v  |-  ( E. x E. y ( x  =  A  /\  y  =  B  /\  ph )  <->  ch )
Distinct variable groups:    x, y, A    x, B, y    ps, x    ch, y
Allowed substitution hints:    ph( x, y)    ps( y)    ch( x)

Proof of Theorem ceqsex2v
StepHypRef Expression
1 nfv 1629 . 2  |-  F/ x ps
2 nfv 1629 . 2  |-  F/ y ch
3 ceqsex2v.1 . 2  |-  A  e. 
_V
4 ceqsex2v.2 . 2  |-  B  e. 
_V
5 ceqsex2v.3 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
6 ceqsex2v.4 . 2  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
71, 2, 3, 4, 5, 6ceqsex2 2992 1  |-  ( E. x E. y ( x  =  A  /\  y  =  B  /\  ph )  <->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2956
This theorem is referenced by:  ceqsex3v  2994  ceqsex4v  2995  ispos  14404  elfuns  25760  brimg  25782  brapply  25783  brsuccf  25786  brrestrict  25794  dfrdg4  25795  diblsmopel  31969
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-v 2958
  Copyright terms: Public domain W3C validator