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Theorem sbc5 2764
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 2749 . 2 ([A / x]φA V)
2 exsimpl 1490 . . 3 (x(x = A φ) → x x = A)
3 isset 2539 . . 3 (A V ↔ x x = A)
42, 3sylibr 137 . 2 (x(x = A φ) → A V)
5 dfsbcq2 2744 . . 3 (y = A → ([y / x]φ[A / x]φ))
6 eqeq2 2031 . . . . 5 (y = A → (x = yx = A))
76anbi1d 441 . . . 4 (y = A → ((x = y φ) ↔ (x = A φ)))
87exbidv 1688 . . 3 (y = A → (x(x = y φ) ↔ x(x = A φ)))
9 sb5 1749 . . 3 ([y / x]φx(x = y φ))
105, 8, 9vtoclbg 2591 . 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   = wceq 1228  wex 1362   wcel 1374  [wsb 1627  Vcvv 2535  [wsbc 2741
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 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-sbc 2742
This theorem is referenced by:  sbc6g  2765  sbc7  2767  sbciegft  2770  sbccomlem  2809  csb2  2831  rexsns  3383  rexsnsOLD  3384
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