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Theorem sbcan 3017
Description: Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.)
Assertion
Ref Expression
sbcan ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))

Proof of Theorem sbcan
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sbcex 2983 . 2 ([𝐴 / 𝑥](𝜑𝜓) → 𝐴 ∈ V)
2 sbcex 2983 . . 3 ([𝐴 / 𝑥]𝜓𝐴 ∈ V)
32adantl 277 . 2 (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) → 𝐴 ∈ V)
4 dfsbcq2 2977 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑𝜓) ↔ [𝐴 / 𝑥](𝜑𝜓)))
5 dfsbcq2 2977 . . . 4 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
6 dfsbcq2 2977 . . . 4 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓[𝐴 / 𝑥]𝜓))
75, 6anbi12d 473 . . 3 (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
8 sban 1965 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
94, 7, 8vtoclbg 2810 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
101, 3, 9pm5.21nii 705 1 ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1363  [wsb 1772  wcel 2158  Vcvv 2749  [wsbc 2974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-sbc 2975
This theorem is referenced by:  sbc3an  3036  difopab  4772  sbcfung  5252  sbcfng  5375  sbcfg  5376  f1od2  6249
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