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Mirrors > Home > MPE Home > Th. List > sbim | Structured version Visualization version GIF version |
Description: Implication inside and outside of 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 2420 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | |
2 | sbi2 2421 | . 2 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓)) | |
3 | 1, 2 | impbii 199 | 1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 [wsb 1937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-10 2059 ax-12 2087 ax-13 2282 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-ex 1745 df-nf 1750 df-sb 1938 |
This theorem is referenced by: sbrim 2424 sblim 2425 sbor 2426 sban 2427 sbbi 2429 sbequ8ALT 2435 sbcimg 3510 mo5f 29452 iuninc 29505 suppss2f 29567 esumpfinvalf 30266 bj-sbnf 32953 wl-sbrimt 33461 wl-sblimt 33462 frege58bcor 38514 frege60b 38516 frege65b 38521 ellimcabssub0 40167 |
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