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Theorem sbc5 2763
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]φx(x = A φ))
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem sbc5
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 sbcex 2748 . 2 ([A / x]φA V)
2 exsimpl 1492 . . 3 (x(x = A φ) → x x = A)
3 isset 2538 . . 3 (A V ↔ x x = A)
42, 3sylibr 137 . 2 (x(x = A φ) → A V)
5 dfsbcq2 2743 . . 3 (y = A → ([y / x]φ[A / x]φ))
6 eqeq2 2032 . . . . 5 (y = A → (x = yx = A))
76anbi1d 441 . . . 4 (y = A → ((x = y φ) ↔ (x = A φ)))
87exbidv 1689 . . 3 (y = A → (x(x = y φ) ↔ x(x = A φ)))
9 sb5 1751 . . 3 ([y / x]φx(x = y φ))
105, 8, 9vtoclbg 2590 . 2 (A V → ([A / x]φx(x = A φ)))
111, 4, 10pm5.21nii 607 1 ([A / x]φx(x = A φ))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98  wex 1362   = wceq 1374   wcel 1376  [wsb 1628  Vcvv 2534  [wsbc 2740
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1378  ax-10 1379  ax-11 1380  ax-i12 1381  ax-bnd 1382  ax-4 1383  ax-17 1402  ax-i9 1406  ax-ial 1411  ax-i5r 1412  ax-ext 2005
This theorem depends on definitions:  df-bi 110  df-tru 1232  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-v 2536  df-sbc 2741
This theorem is referenced by:  sbc6g  2764  sbc7  2766  sbciegft  2769  sbccomlem  2809  csb2  2831  rexsns  3361  rexsnsOLD  3362
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