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Mirrors > Home > ILE Home > Th. List > sbcor | GIF version |
Description: Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.) |
Ref | Expression |
---|---|
sbcor | ⊢ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 2823 | . 2 ⊢ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) → 𝐴 ∈ V) | |
2 | sbcex 2823 | . . 3 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) | |
3 | sbcex 2823 | . . 3 ⊢ ([𝐴 / 𝑥]𝜓 → 𝐴 ∈ V) | |
4 | 2, 3 | jaoi 668 | . 2 ⊢ (([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓) → 𝐴 ∈ V) |
5 | dfsbcq2 2818 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑 ∨ 𝜓) ↔ [𝐴 / 𝑥](𝜑 ∨ 𝜓))) | |
6 | dfsbcq2 2818 | . . . 4 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
7 | dfsbcq2 2818 | . . . 4 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜓)) | |
8 | 6, 7 | orbi12d 739 | . . 3 ⊢ (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓))) |
9 | sbor 1869 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓)) | |
10 | 5, 8, 9 | vtoclbg 2659 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓))) |
11 | 1, 4, 10 | pm5.21nii 652 | 1 ⊢ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 ∨ wo 661 = wceq 1284 ∈ wcel 1433 [wsb 1685 Vcvv 2601 [wsbc 2815 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-sbc 2816 |
This theorem is referenced by: rabrsndc 3460 |
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