ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbc5 Unicode version

Theorem sbc5 2810
Description: An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.)
Assertion
Ref Expression
sbc5  |-  ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem sbc5
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 sbcex 2795 . 2  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
2 exsimpl 1524 . . 3  |-  ( E. x ( x  =  A  /\  ph )  ->  E. x  x  =  A )
3 isset 2578 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
42, 3sylibr 141 . 2  |-  ( E. x ( x  =  A  /\  ph )  ->  A  e.  _V )
5 dfsbcq2 2790 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
6 eqeq2 2065 . . . . 5  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
76anbi1d 446 . . . 4  |-  ( y  =  A  ->  (
( x  =  y  /\  ph )  <->  ( x  =  A  /\  ph )
) )
87exbidv 1722 . . 3  |-  ( y  =  A  ->  ( E. x ( x  =  y  /\  ph )  <->  E. x ( x  =  A  /\  ph )
) )
9 sb5 1783 . . 3  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
105, 8, 9vtoclbg 2631 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ph  <->  E. x ( x  =  A  /\  ph ) ) )
111, 4, 10pm5.21nii 630 1  |-  ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102    = wceq 1259   E.wex 1397    e. wcel 1409   [wsb 1661   _Vcvv 2574   [.wsbc 2787
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  df-sbc 2788
This theorem is referenced by:  sbc6g  2811  sbc7  2813  sbciegft  2816  sbccomlem  2860  csb2  2882  rexsns  3437  rexsnsOLD  3438
  Copyright terms: Public domain W3C validator