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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 3798 | . 2 ⊢ ([𝐴 / 𝑥](𝜑 ∧ 𝜓) → 𝐴 ∈ V) | |
| 2 | sbcex 3798 | . . 3 ⊢ ([𝐴 / 𝑥]𝜓 → 𝐴 ∈ V) | |
| 3 | 2 | adantl 481 | . 2 ⊢ (([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓) → 𝐴 ∈ V) |
| 4 | dfsbcq2 3791 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ [𝐴 / 𝑥](𝜑 ∧ 𝜓))) | |
| 5 | dfsbcq2 3791 | . . . 4 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
| 6 | dfsbcq2 3791 | . . . 4 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜓)) | |
| 7 | 5, 6 | anbi12d 632 | . . 3 ⊢ (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓))) |
| 8 | sban 2080 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓)) | |
| 9 | 4, 7, 8 | vtoclbg 3557 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓))) |
| 10 | 1, 3, 9 | pm5.21nii 378 | 1 ⊢ ([𝐴 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 [wsb 2064 ∈ wcel 2108 Vcvv 3480 [wsbc 3788 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-sbc 3789 |
| This theorem is referenced by: sbc3an 3855 sbcabel 3878 2nreu 4444 csbopg 4891 csbuni 4936 csbmpt12 5562 csbxp 5785 difopabOLD 5841 sbcfung 6590 sbcfng 6733 sbcfg 6734 fmptsnd 7189 csbfrecsg 8309 f1od2 32732 esum2dlem 34093 bnj976 34791 bnj110 34872 bnj1040 34986 csboprabg 37331 csbmpo123 37332 f1omptsnlem 37337 mptsnunlem 37339 relowlpssretop 37365 csbfinxpg 37389 sbcani 38115 sbccom2lem 38131 minregex 43547 brtrclfv2 43740 cotrclrcl 43755 frege124d 43774 sbiota1 44453 onfrALTlem5 44562 onfrALTlem4 44563 csbingVD 44904 onfrALTlem5VD 44905 onfrALTlem4VD 44906 csbxpgVD 44914 csbunigVD 44918 rspesbcd 44958 |
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