Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-sblimt | Structured version Visualization version GIF version |
Description: Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2305. (Contributed by Wolf Lammen, 26-Jul-2019.) |
Ref | Expression |
---|---|
wl-sblimt | ⊢ (Ⅎ𝑥𝜓 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbim 2304 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | |
2 | sbft 2266 | . . 3 ⊢ (Ⅎ𝑥𝜓 → ([𝑦 / 𝑥]𝜓 ↔ 𝜓)) | |
3 | 2 | imbi2d 344 | . 2 ⊢ (Ⅎ𝑥𝜓 → (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓))) |
4 | 1, 3 | syl5bb 286 | 1 ⊢ (Ⅎ𝑥𝜓 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 Ⅎwnf 1791 [wsb 2070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-10 2141 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ex 1788 df-nf 1792 df-sb 2071 |
This theorem is referenced by: (None) |
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