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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
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem sbc5
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sbcex 3162 . 2
2 exsimpl 1602 . . 3
3 isset 2952 . . 3
42, 3sylibr 204 . 2
5 dfsbcq2 3156 . . 3
6 eqeq2 2444 . . . . 5
76anbi1d 686 . . . 4
87exbidv 1636 . . 3
9 sb5 2175 . . 3
105, 8, 9vtoclbg 3004 . 2
111, 4, 10pm5.21nii 343 1
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359  wex 1550   wceq 1652  wsb 1658   wcel 1725  cvv 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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