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Mirrors > Home > MPE Home > Th. List > sbcimg | Structured version Visualization version GIF version |
Description: Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.) |
Ref | Expression |
---|---|
sbcimg | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq2 3746 | . 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ [𝐴 / 𝑥](𝜑 → 𝜓))) | |
2 | dfsbcq2 3746 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
3 | dfsbcq2 3746 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜓)) | |
4 | 2, 3 | imbi12d 345 | . 2 ⊢ (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))) |
5 | sbim 2300 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | |
6 | 1, 4, 5 | vtoclbg 3530 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 [wsb 2068 ∈ wcel 2107 [wsbc 3743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-sbc 3744 |
This theorem is referenced by: sbcim1OLD 3800 sbceqalOLD 3810 sbc19.21g 3821 sbcssg 4485 iota4an 6482 sbcfung 6529 riotass2 7348 tfinds2 7804 telgsums 19778 bnj110 33534 bnj92 33538 bnj539 33567 bnj540 33568 f1omptsnlem 35857 mptsnunlem 35859 topdifinffinlem 35868 relowlpssretop 35885 rdgeqoa 35891 sbcimi 36619 cdlemkid3N 39446 cdlemkid4 39447 cdlemk35s 39450 cdlemk39s 39452 cdlemk42 39454 frege77 42304 frege116 42343 frege118 42345 sbcim2g 42912 onfrALTlem5 42916 sbcim2gVD 43249 sbcssgVD 43257 onfrALTlem5VD 43259 iccelpart 45715 |
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