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| 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 2073 | . . 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 2073 | . 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 2063 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-10 2140 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-nf 1783 df-sb 2064 | 
| This theorem is referenced by: sbim 2302 | 
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