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Mirrors > Home > MPE Home > Th. List > sbi2 | Structured version Visualization version GIF version |
Description: Introduction of implication into substitution. (Contributed by NM, 14-May-1993.) |
Ref | Expression |
---|---|
sbi2 | ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbn 2279 | . . 3 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑) | |
2 | pm2.21 123 | . . . 4 ⊢ (¬ 𝜑 → (𝜑 → 𝜓)) | |
3 | 2 | sbimi 2072 | . . 3 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 → [𝑦 / 𝑥](𝜑 → 𝜓)) |
4 | 1, 3 | sylbir 235 | . 2 ⊢ (¬ [𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜑 → 𝜓)) |
5 | ax-1 6 | . . 3 ⊢ (𝜓 → (𝜑 → 𝜓)) | |
6 | 5 | sbimi 2072 | . 2 ⊢ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥](𝜑 → 𝜓)) |
7 | 4, 6 | ja 186 | 1 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 [wsb 2062 |
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 1908 ax-6 1965 ax-7 2005 ax-10 2139 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-nf 1781 df-sb 2063 |
This theorem is referenced by: sbim 2302 |
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