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Theorem sbcan 2782
 Description: Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.)
Assertion
Ref Expression
sbcan ([A / x](φ ψ) ↔ ([A / x]φ [A / x]ψ))

Proof of Theorem sbcan
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 sbcex 2749 . 2 ([A / x](φ ψ) → A V)
2 sbcex 2749 . . 3 ([A / x]ψA V)
32adantl 262 . 2 (([A / x]φ [A / x]ψ) → A V)
4 dfsbcq2 2744 . . 3 (y = A → ([y / x](φ ψ) ↔ [A / x](φ ψ)))
5 dfsbcq2 2744 . . . 4 (y = A → ([y / x]φ[A / x]φ))
6 dfsbcq2 2744 . . . 4 (y = A → ([y / x]ψ[A / x]ψ))
75, 6anbi12d 445 . . 3 (y = A → (([y / x]φ [y / x]ψ) ↔ ([A / x]φ [A / x]ψ)))
8 sban 1811 . . 3 ([y / x](φ ψ) ↔ ([y / x]φ [y / x]ψ))
94, 7, 8vtoclbg 2591 . 2 (A V → ([A / x](φ ψ) ↔ ([A / x]φ [A / x]ψ)))
101, 3, 9pm5.21nii 607 1 ([A / x](φ ψ) ↔ ([A / x]φ [A / x]ψ))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1228   ∈ 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:  difopab  4396  sbcfung  4851  sbcfng  4970  sbcfg  4971
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