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Mirrors > Home > MPE Home > Th. List > sbcbig | Structured version Visualization version GIF version |
Description: Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.) |
Ref | Expression |
---|---|
sbcbig | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq2 3777 | . 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ [𝐴 / 𝑥](𝜑 ↔ 𝜓))) | |
2 | dfsbcq2 3777 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
3 | dfsbcq2 3777 | . . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜓)) | |
4 | 2, 3 | bibi12d 348 | . 2 ⊢ (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) |
5 | sbbi 2317 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓)) | |
6 | 1, 4, 5 | vtoclbg 3571 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 [wsb 2069 ∈ wcel 2114 [wsbc 3774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-sbc 3775 |
This theorem is referenced by: sbcbi1 3832 sbcabel 3863 opsbc2ie 30241 bnj89 31993 bj-sbeq 34220 bj-sbceqgALT 34221 sbcbi 40880 sbc3orgVD 41192 sbcbiVD 41217 |
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