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Theorem sbc5 2781
 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 2766 . 2 ([A / x]φA V)
2 exsimpl 1505 . . 3 (x(x = A φ) → x x = A)
3 isset 2555 . . 3 (A V ↔ x x = A)
42, 3sylibr 137 . 2 (x(x = A φ) → A V)
5 dfsbcq2 2761 . . 3 (y = A → ([y / x]φ[A / x]φ))
6 eqeq2 2046 . . . . 5 (y = A → (x = yx = A))
76anbi1d 438 . . . 4 (y = A → ((x = y φ) ↔ (x = A φ)))
87exbidv 1703 . . 3 (y = A → (x(x = y φ) ↔ x(x = A φ)))
9 sb5 1764 . . 3 ([y / x]φx(x = y φ))
105, 8, 9vtoclbg 2608 . 2 (A V → ([A / x]φx(x = A φ)))
111, 4, 10pm5.21nii 619 1 ([A / x]φx(x = A φ))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  [wsb 1642  Vcvv 2551  [wsbc 2758 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759 This theorem is referenced by:  sbc6g  2782  sbc7  2784  sbciegft  2787  sbccomlem  2826  csb2  2848  rexsns  3400  rexsnsOLD  3401
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