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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 3683 | . 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ [𝐴 / 𝑥](𝜑 → 𝜓))) | |
2 | dfsbcq2 3683 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
3 | dfsbcq2 3683 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜓)) | |
4 | 2, 3 | imbi12d 348 | . 2 ⊢ (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))) |
5 | sbim 2307 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | |
6 | 1, 4, 5 | vtoclbg 3472 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1542 [wsb 2074 ∈ wcel 2114 [wsbc 3680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-12 2179 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-sbc 3681 |
This theorem is referenced by: sbcim1 3734 sbceqal 3744 sbc19.21g 3754 sbcssg 4410 iota4an 6321 sbcfung 6363 riotass2 7158 tfinds2 7597 telgsums 19232 bnj110 32409 bnj92 32413 bnj539 32442 bnj540 32443 f1omptsnlem 35130 mptsnunlem 35132 topdifinffinlem 35141 relowlpssretop 35158 rdgeqoa 35164 sbcimi 35891 cdlemkid3N 38570 cdlemkid4 38571 cdlemk35s 38574 cdlemk39s 38576 cdlemk42 38578 frege77 41094 frege116 41133 frege118 41135 sbcim2g 41696 onfrALTlem5 41700 sbcim2gVD 42033 sbcssgVD 42041 onfrALTlem5VD 42043 iccelpart 44419 |
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