![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > sbim | Structured version Visualization version GIF version |
Description: Implication inside and outside of a substitution are equivalent. For a version requiring disjoint variables, but fewer axioms, see sbimv 2280. (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 2468 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | |
2 | sbi2 2469 | . 2 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓)) | |
3 | 1, 2 | impbii 201 | 1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 [wsb 2011 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-10 2135 ax-12 2163 ax-13 2334 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-ex 1824 df-nf 1828 df-sb 2012 |
This theorem is referenced by: sbrim 2472 sblim 2473 sbor 2474 sban 2475 sbbi 2477 sbequ8ALT 2483 sbcimg 3695 mo5f 29900 iuninc 29945 suppss2f 30008 esumpfinvalf 30740 bj-sbnf 33407 wl-sbrimt 33929 wl-sblimt 33930 frege58bcor 39163 frege60b 39165 frege65b 39170 ellimcabssub0 40767 |
Copyright terms: Public domain | W3C validator |