![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 3595 | . 2 ⊢ ([𝐴 / 𝑥](𝜑 ∧ 𝜓) → 𝐴 ∈ V) | |
2 | sbcex 3595 | . . 3 ⊢ ([𝐴 / 𝑥]𝜓 → 𝐴 ∈ V) | |
3 | 2 | adantl 467 | . 2 ⊢ (([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓) → 𝐴 ∈ V) |
4 | dfsbcq2 3588 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ [𝐴 / 𝑥](𝜑 ∧ 𝜓))) | |
5 | dfsbcq2 3588 | . . . 4 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
6 | dfsbcq2 3588 | . . . 4 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜓)) | |
7 | 5, 6 | anbi12d 608 | . . 3 ⊢ (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓))) |
8 | sban 2545 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓)) | |
9 | 4, 7, 8 | vtoclbg 3416 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓))) |
10 | 1, 3, 9 | pm5.21nii 367 | 1 ⊢ ([𝐴 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 382 = wceq 1630 [wsb 2048 ∈ wcel 2144 Vcvv 3349 [wsbc 3585 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-9 2153 ax-10 2173 ax-12 2202 ax-13 2407 ax-ext 2750 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-clab 2757 df-cleq 2763 df-clel 2766 df-v 3351 df-sbc 3586 |
This theorem is referenced by: sbc3an 3643 sbcabel 3664 csbopg 4555 csbuni 4600 csbmpt12 5143 csbxp 5340 difopab 5392 sbcfung 6055 sbcfng 6182 sbcfg 6183 fmptsnd 6578 f1od2 29833 esum2dlem 30488 bnj976 31180 bnj110 31260 bnj1040 31372 csbwrecsg 33503 csboprabg 33506 csbmpt22g 33507 f1omptsnlem 33513 mptsnunlem 33515 relowlpssretop 33542 csbfinxpg 33555 sbcani 34235 sbccom2lem 34254 brtrclfv2 38538 cotrclrcl 38553 frege124d 38572 sbiota1 39154 onfrALTlem5 39276 onfrALTlem4 39277 csbxpgOLD 39570 onfrALTlem5VD 39637 |
Copyright terms: Public domain | W3C validator |