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Theorem sbc5 3177
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 3162 . 2  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
2 exsimpl 1602 . . 3  |-  ( E. x ( x  =  A  /\  ph )  ->  E. x  x  =  A )
3 isset 2952 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
42, 3sylibr 204 . 2  |-  ( E. x ( x  =  A  /\  ph )  ->  A  e.  _V )
5 dfsbcq2 3156 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
6 eqeq2 2444 . . . . 5  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
76anbi1d 686 . . . 4  |-  ( y  =  A  ->  (
( x  =  y  /\  ph )  <->  ( x  =  A  /\  ph )
) )
87exbidv 1636 . . 3  |-  ( y  =  A  ->  ( E. x ( x  =  y  /\  ph )  <->  E. x ( x  =  A  /\  ph )
) )
9 sb5 2175 . . 3  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
105, 8, 9vtoclbg 3004 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ph  <->  E. x ( x  =  A  /\  ph ) ) )
111, 4, 10pm5.21nii 343 1  |-  ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652   [wsb 1658    e. wcel 1725   _Vcvv 2948   [.wsbc 3153
This theorem is referenced by:  sbc6g  3178  sbc7  3180  sbciegft  3183  sbccomlem  3223  csb2  3245  rexsns  3837  pm13.192  27542  pm13.195  27545  2sbc5g  27548  iotasbc  27551  pm14.122b  27555  iotasbc5  27563
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-sbc 3154
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