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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 2076 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | |
2 | sbi2 2310 | . 2 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓)) | |
3 | 1, 2 | impbii 211 | 1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 |
This theorem is referenced by: sbanOLD 2312 sbrim 2313 sblim 2315 sbor 2316 sbbi 2317 sbequ8 2543 sbcimg 3820 mo5f 30253 iuninc 30312 suppss2f 30384 esumpfinvalf 31335 bj-sbnf 34164 wl-sbrimt 34801 wl-sblimt 34802 wl-dfrmosb 34868 frege58bcor 40269 frege60b 40271 frege65b 40276 ellimcabssub0 41918 |
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