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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 3781 | . 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ [𝐴 / 𝑥](𝜑 → 𝜓))) | |
2 | dfsbcq2 3781 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
3 | dfsbcq2 3781 | . . 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 3560 | 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 3778 |
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 3779 |
This theorem is referenced by: sbcim1OLD 3835 sbceqalOLD 3845 sbc19.21g 3856 sbcssg 4524 iota4an 6526 sbcfung 6573 riotass2 7396 tfinds2 7853 telgsums 19861 bnj110 33900 bnj92 33904 bnj539 33933 bnj540 33934 f1omptsnlem 36265 mptsnunlem 36267 topdifinffinlem 36276 relowlpssretop 36293 rdgeqoa 36299 sbcimi 37026 cdlemkid3N 39852 cdlemkid4 39853 cdlemk35s 39856 cdlemk39s 39858 cdlemk42 39860 frege77 42739 frege116 42778 frege118 42780 sbcim2g 43347 onfrALTlem5 43351 sbcim2gVD 43684 sbcssgVD 43692 onfrALTlem5VD 43694 iccelpart 46149 |
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