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Theorem sbc5 3361
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 3346 . 2  |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
2 exsimpl 1654 . . 3  |-  ( E. x ( x  =  A  /\  ph )  ->  E. x  x  =  A )
3 isset 3122 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
42, 3sylibr 212 . 2  |-  ( E. x ( x  =  A  /\  ph )  ->  A  e.  _V )
5 dfsbcq2 3339 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
6 eqeq2 2482 . . . . 5  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
76anbi1d 704 . . . 4  |-  ( y  =  A  ->  (
( x  =  y  /\  ph )  <->  ( x  =  A  /\  ph )
) )
87exbidv 1690 . . 3  |-  ( y  =  A  ->  ( E. x ( x  =  y  /\  ph )  <->  E. x ( x  =  A  /\  ph )
) )
9 sb5 2157 . . 3  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
105, 8, 9vtoclbg 3177 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ph  <->  E. x ( x  =  A  /\  ph ) ) )
111, 4, 10pm5.21nii 353 1  |-  ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596   [wsb 1711    e. wcel 1767   _Vcvv 3118   [.wsbc 3336
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-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-v 3120  df-sbc 3337
This theorem is referenced by:  sbc6g  3362  sbc7  3364  sbciegft  3367  sbccomlem  3415  csb2  3442  rexsns  4066  rexsnsOLD  4067  sbccom2lem  30457  pm13.192  31219  pm13.195  31222  2sbc5g  31225  iotasbc  31228  pm14.122b  31232  iotasbc5  31240
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