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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 3716 | . 2 ⊢ ([𝐴 / 𝑥](𝜑 ∧ 𝜓) → 𝐴 ∈ V) | |
2 | sbcex 3716 | . . 3 ⊢ ([𝐴 / 𝑥]𝜓 → 𝐴 ∈ V) | |
3 | 2 | adantl 482 | . 2 ⊢ (([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓) → 𝐴 ∈ V) |
4 | dfsbcq2 3709 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ [𝐴 / 𝑥](𝜑 ∧ 𝜓))) | |
5 | dfsbcq2 3709 | . . . 4 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
6 | dfsbcq2 3709 | . . . 4 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜓)) | |
7 | 5, 6 | anbi12d 630 | . . 3 ⊢ (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓))) |
8 | sban 2059 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓)) | |
9 | 4, 7, 8 | vtoclbg 3511 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓))) |
10 | 1, 3, 9 | pm5.21nii 380 | 1 ⊢ ([𝐴 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1522 [wsb 2042 ∈ wcel 2081 Vcvv 3437 [wsbc 3706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-v 3439 df-sbc 3707 |
This theorem is referenced by: sbc3an 3766 sbcabel 3789 2nreu 4307 csbopg 4728 csbuni 4773 csbmpt12 5332 csbxp 5536 difopab 5588 sbcfung 6249 sbcfng 6379 sbcfg 6380 fmptsnd 6794 f1od2 30145 esum2dlem 30968 bnj976 31666 bnj110 31746 bnj1040 31858 csbwrecsg 34139 csboprabg 34142 csbmpo123 34143 f1omptsnlem 34148 mptsnunlem 34150 relowlpssretop 34176 csbfinxpg 34200 sbcani 34918 sbccom2lem 34934 brtrclfv2 39557 cotrclrcl 39572 frege124d 39591 sbiota1 40304 onfrALTlem5 40415 onfrALTlem4 40416 csbingVD 40757 onfrALTlem5VD 40758 onfrALTlem4VD 40759 csbxpgVD 40767 csbunigVD 40771 |
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