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

Theorem equsexh 2142
Description: An equivalence related to implicit substitution. See equsexhv 2077 for a version with a dv condition which does not require ax-13 2101. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
equsexh.1  |-  ( ps 
->  A. x ps )
equsexh.2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
equsexh  |-  ( E. x ( x  =  y  /\  ph )  <->  ps )

Proof of Theorem equsexh
StepHypRef Expression
1 equsexh.1 . . 3  |-  ( ps 
->  A. x ps )
21nfi 1684 . 2  |-  F/ x ps
3 equsexh.2 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
42, 3equsex 2140 1  |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375   A.wal 1452   E.wex 1673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943  ax-13 2101
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-nf 1678
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator