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

Theorem ceqsal 3140
Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)
Hypotheses
Ref Expression
ceqsal.1  |-  F/ x ps
ceqsal.2  |-  A  e. 
_V
ceqsal.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ceqsal  |-  ( A. x ( x  =  A  ->  ph )  <->  ps )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem ceqsal
StepHypRef Expression
1 ceqsal.2 . 2  |-  A  e. 
_V
2 ceqsal.1 . . 3  |-  F/ x ps
3 ceqsal.3 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
42, 3ceqsalg 3138 . 2  |-  ( A  e.  _V  ->  ( A. x ( x  =  A  ->  ph )  <->  ps )
)
51, 4ax-mp 5 1  |-  ( A. x ( x  =  A  ->  ph )  <->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1377    = wceq 1379   F/wnf 1599    e. wcel 1767   _Vcvv 3113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-v 3115
This theorem is referenced by:  ceqsalv  3141  aomclem6  30609
  Copyright terms: Public domain W3C validator