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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 3444 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 3412 | . 2 ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → 𝐴 ∈ V) | |
2 | sbcimg 3444 | . . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))) | |
3 | 2 | biimpd 218 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))) |
4 | 1, 3 | mpcom 37 | 1 ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 Vcvv 3173 [wsbc 3402 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-v 3175 df-sbc 3403 |
This theorem is referenced by: sbcimdv 3465 sbcimdvOLD 3466 frege59c 37236 frege60c 37237 frege62c 37239 frege65c 37242 frege70 37247 frege72 37249 frege92 37269 frege120 37297 |
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