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Theorem sbcor 3845
Description: Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 17-Aug-2018.)
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 3801 . 2 ([𝐴 / 𝑥](𝜑𝜓) → 𝐴 ∈ V)
2 sbcex 3801 . . 3 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
3 sbcex 3801 . . 3 ([𝐴 / 𝑥]𝜓𝐴 ∈ V)
42, 3jaoi 857 . 2 (([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓) → 𝐴 ∈ V)
5 dfsbcq2 3794 . . 3 (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑𝜓) ↔ [𝐴 / 𝑥](𝜑𝜓)))
6 dfsbcq2 3794 . . . 4 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
7 dfsbcq2 3794 . . . 4 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓[𝐴 / 𝑥]𝜓))
86, 7orbi12d 918 . . 3 (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
9 sbor 2306 . . 3 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓))
105, 8, 9vtoclbg 3557 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))
111, 4, 10pm5.21nii 378 1 ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wo 847   = wceq 1537  [wsb 2062  wcel 2106  Vcvv 3478  [wsbc 3791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-sbc 3792
This theorem is referenced by:  sbcori  38096  sbc3or  44530  sbc3orgVD  44849  sbcoreleleqVD  44857
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