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

Theorem equsexh 1903
Description: A useful equivalence related to substitution. (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 1538 . 2  |-  F/ x ps
3 equsexh.2 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
42, 3equsex 1902 1  |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528
This theorem is referenced by:  cleljust  1954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532
  Copyright terms: Public domain W3C validator