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Theorem sbc5 2986
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 2971 . 2  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
2 exsimpl 1617 . . 3  |-  ( E. x ( x  =  A  /\  ph )  ->  E. x  x  =  A )
3 isset 2743 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
42, 3sylibr 134 . 2  |-  ( E. x ( x  =  A  /\  ph )  ->  A  e.  _V )
5 dfsbcq2 2965 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
6 eqeq2 2187 . . . . 5  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
76anbi1d 465 . . . 4  |-  ( y  =  A  ->  (
( x  =  y  /\  ph )  <->  ( x  =  A  /\  ph )
) )
87exbidv 1825 . . 3  |-  ( y  =  A  ->  ( E. x ( x  =  y  /\  ph )  <->  E. x ( x  =  A  /\  ph )
) )
9 sb5 1887 . . 3  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
105, 8, 9vtoclbg 2798 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ph  <->  E. x ( x  =  A  /\  ph ) ) )
111, 4, 10pm5.21nii 704 1  |-  ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1353   E.wex 1492   [wsb 1762    e. wcel 2148   _Vcvv 2737   [.wsbc 2962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sbc 2963
This theorem is referenced by:  sbc6g  2987  sbc7  2989  sbciegft  2993  sbccomlem  3037  csb2  3059  rexsns  3631
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