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Mirrors > Home > MPE Home > Th. List > sbcim1 | Structured version Visualization version GIF version |
Description: Distribution of class substitution over implication. One direction of sbcimg 3819 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
Ref | Expression |
---|---|
sbcim1 | ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 3781 | . 2 ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → 𝐴 ∈ V) | |
2 | sbcimg 3819 | . . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))) | |
3 | 2 | biimpd 231 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))) |
4 | 1, 3 | mpcom 38 | 1 ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 Vcvv 3494 [wsbc 3771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-v 3496 df-sbc 3772 |
This theorem is referenced by: sbcimdv 3842 opsbc2ie 30238 frege59c 40266 frege60c 40267 frege62c 40269 frege65c 40272 frege70 40277 frege72 40279 frege92 40299 frege120 40327 |
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