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Theorem sbcor 2956
Description: Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.)
Assertion
Ref Expression
sbcor ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))

Proof of Theorem sbcor
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sbcex 2920 . 2 ([𝐴 / 𝑥](𝜑𝜓) → 𝐴 ∈ V)
2 sbcex 2920 . . 3 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
3 sbcex 2920 . . 3 ([𝐴 / 𝑥]𝜓𝐴 ∈ V)
42, 3jaoi 706 . 2 (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) → 𝐴 ∈ V)
5 dfsbcq2 2915 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑𝜓) ↔ [𝐴 / 𝑥](𝜑𝜓)))
6 dfsbcq2 2915 . . . 4 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
7 dfsbcq2 2915 . . . 4 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓[𝐴 / 𝑥]𝜓))
86, 7orbi12d 783 . . 3 (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
9 sbor 1928 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓))
105, 8, 9vtoclbg 2750 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
111, 4, 10pm5.21nii 694 1 ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
Colors of variables: wff set class
Syntax hints:  wb 104  wo 698   = wceq 1332  wcel 1481  [wsb 1736  Vcvv 2689  [wsbc 2912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sbc 2913
This theorem is referenced by:  rabrsndc  3598
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