Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-sbrimt | Structured version Visualization version GIF version |
Description: Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2384. (Contributed by Wolf Lammen, 26-Jul-2019.) |
Ref | Expression |
---|---|
wl-sbrimt | ⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbim 2383 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | |
2 | sbft 2367 | . . 3 ⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑)) | |
3 | 2 | imbi1d 330 | . 2 ⊢ (Ⅎ𝑥𝜑 → (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))) |
4 | 1, 3 | syl5bb 271 | 1 ⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 Ⅎwnf 1699 [wsb 1867 |
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 |
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 |
This theorem is referenced by: (None) |
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