![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > sbcor | Structured version Visualization version GIF version |
Description: Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 17-Aug-2018.) |
Ref | Expression |
---|---|
sbcor | ⊢ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 3783 | . 2 ⊢ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) → 𝐴 ∈ V) | |
2 | sbcex 3783 | . . 3 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) | |
3 | sbcex 3783 | . . 3 ⊢ ([𝐴 / 𝑥]𝜓 → 𝐴 ∈ V) | |
4 | 2, 3 | jaoi 855 | . 2 ⊢ (([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓) → 𝐴 ∈ V) |
5 | dfsbcq2 3776 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑 ∨ 𝜓) ↔ [𝐴 / 𝑥](𝜑 ∨ 𝜓))) | |
6 | dfsbcq2 3776 | . . . 4 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
7 | dfsbcq2 3776 | . . . 4 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜓)) | |
8 | 6, 7 | orbi12d 916 | . . 3 ⊢ (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓))) |
9 | sbor 2296 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓)) | |
10 | 5, 8, 9 | vtoclbg 3535 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓))) |
11 | 1, 4, 10 | pm5.21nii 377 | 1 ⊢ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∨ wo 845 = wceq 1533 [wsb 2059 ∈ wcel 2098 Vcvv 3461 [wsbc 3773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2166 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-v 3463 df-sbc 3774 |
This theorem is referenced by: sbcori 37715 sbc3or 44115 sbc3orgVD 44434 sbcoreleleqVD 44442 |
Copyright terms: Public domain | W3C validator |