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Mirrors > Home > MPE Home > Th. List > sbim | Structured version Visualization version GIF version |
Description: Implication inside and outside of a substitution are equivalent. (Contributed by NM, 14-May-1993.) |
Ref | Expression |
---|---|
sbim | ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbi1 2079 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | |
2 | sbi2 2305 | . 2 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓)) | |
3 | 1, 2 | impbii 212 | 1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 [wsb 2072 |
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 2018 ax-10 2143 ax-12 2177 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ex 1788 df-nf 1792 df-sb 2073 |
This theorem is referenced by: sbrim 2307 sblim 2309 sbor 2310 sbbi 2311 sbequ8 2504 sbcimg 3734 mo5f 30510 iuninc 30573 suppss2f 30647 esumpfinvalf 31710 bj-sbnf 34710 wl-sbrimt 35391 wl-sblimt 35392 frege58bcor 41129 frege60b 41131 frege65b 41136 ellimcabssub0 42776 |
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