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Mirrors > Home > MPE Home > Th. List > sblim | Structured version Visualization version GIF version |
Description: Substitution in an implication with a variable not free in the consequent affects only the antecedent. (Contributed by NM, 14-Nov-2013.) (Revised by Mario Carneiro, 4-Oct-2016.) |
Ref | Expression |
---|---|
sblim.1 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
sblim | ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbim 2307 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | |
2 | sblim.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
3 | 2 | sbf 2266 | . . 3 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜓) |
4 | 3 | imbi2i 338 | . 2 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓)) |
5 | 1, 4 | bitri 277 | 1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 Ⅎwnf 1780 [wsb 2065 |
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-10 2141 ax-12 2172 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1777 df-nf 1781 df-sb 2066 |
This theorem is referenced by: sbmo 2694 |
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