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Mirrors > Home > MPE Home > Th. List > sbcan | Structured version Visualization version GIF version |
Description: Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 17-Aug-2018.) |
Ref | Expression |
---|---|
sbcan | ⊢ ([𝐴 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 3786 | . 2 ⊢ ([𝐴 / 𝑥](𝜑 ∧ 𝜓) → 𝐴 ∈ V) | |
2 | sbcex 3786 | . . 3 ⊢ ([𝐴 / 𝑥]𝜓 → 𝐴 ∈ V) | |
3 | 2 | adantl 482 | . 2 ⊢ (([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓) → 𝐴 ∈ V) |
4 | dfsbcq2 3779 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ [𝐴 / 𝑥](𝜑 ∧ 𝜓))) | |
5 | dfsbcq2 3779 | . . . 4 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
6 | dfsbcq2 3779 | . . . 4 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜓)) | |
7 | 5, 6 | anbi12d 631 | . . 3 ⊢ (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓))) |
8 | sban 2083 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓)) | |
9 | 4, 7, 8 | vtoclbg 3559 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓))) |
10 | 1, 3, 9 | pm5.21nii 379 | 1 ⊢ ([𝐴 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 [wsb 2067 ∈ wcel 2106 Vcvv 3474 [wsbc 3776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-sbc 3777 |
This theorem is referenced by: sbc3an 3846 sbcabel 3871 2nreu 4440 csbopg 4890 csbuni 4939 csbmpt12 5556 csbxp 5773 difopabOLD 5829 sbcfung 6569 sbcfng 6711 sbcfg 6712 fmptsnd 7163 csbfrecsg 8265 f1od2 31933 esum2dlem 33078 bnj976 33776 bnj110 33857 bnj1040 33971 csboprabg 36199 csbmpo123 36200 f1omptsnlem 36205 mptsnunlem 36207 relowlpssretop 36233 csbfinxpg 36257 sbcani 36964 sbccom2lem 36980 minregex 42270 brtrclfv2 42463 cotrclrcl 42478 frege124d 42497 sbiota1 43178 onfrALTlem5 43288 onfrALTlem4 43289 csbingVD 43630 onfrALTlem5VD 43631 onfrALTlem4VD 43632 csbxpgVD 43640 csbunigVD 43644 |
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