ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ceqsexg Unicode version

Theorem ceqsexg 2695
Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004.)
Hypotheses
Ref Expression
ceqsexg.1  |-  F/ x ps
ceqsexg.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ceqsexg  |-  ( A  e.  V  ->  ( E. x ( x  =  A  /\  ph )  <->  ps ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)    V( x)

Proof of Theorem ceqsexg
StepHypRef Expression
1 nfcv 2194 . 2  |-  F/_ x A
2 nfe1 1401 . . 3  |-  F/ x E. x ( x  =  A  /\  ph )
3 ceqsexg.1 . . 3  |-  F/ x ps
42, 3nfbi 1497 . 2  |-  F/ x
( E. x ( x  =  A  /\  ph )  <->  ps )
5 ceqex 2694 . . 3  |-  ( x  =  A  ->  ( ph 
<->  E. x ( x  =  A  /\  ph ) ) )
6 ceqsexg.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
75, 6bibi12d 228 . 2  |-  ( x  =  A  ->  (
( ph  <->  ph )  <->  ( E. x ( x  =  A  /\  ph )  <->  ps ) ) )
8 biid 164 . 2  |-  ( ph  <->  ph )
91, 4, 7, 8vtoclgf 2629 1  |-  ( A  e.  V  ->  ( E. x ( x  =  A  /\  ph )  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259   F/wnf 1365   E.wex 1397    e. wcel 1409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576
This theorem is referenced by:  ceqsexgv  2696
  Copyright terms: Public domain W3C validator