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Mirrors > Home > MPE Home > Th. List > Mathboxes > ifpim1 | Structured version Visualization version GIF version |
Description: Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
Ref | Expression |
---|---|
ifpim1 | ⊢ ((𝜑 → 𝜓) ↔ if-(¬ 𝜑, ⊤, 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tru 1479 | . . . 4 ⊢ ⊤ | |
2 | 1 | olci 405 | . . 3 ⊢ (¬ ¬ 𝜑 ∨ ⊤) |
3 | 2 | biantrur 526 | . 2 ⊢ ((¬ 𝜑 ∨ 𝜓) ↔ ((¬ ¬ 𝜑 ∨ ⊤) ∧ (¬ 𝜑 ∨ 𝜓))) |
4 | imor 427 | . 2 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓)) | |
5 | dfifp4 1010 | . 2 ⊢ (if-(¬ 𝜑, ⊤, 𝜓) ↔ ((¬ ¬ 𝜑 ∨ ⊤) ∧ (¬ 𝜑 ∨ 𝜓))) | |
6 | 3, 4, 5 | 3bitr4i 291 | 1 ⊢ ((𝜑 → 𝜓) ↔ if-(¬ 𝜑, ⊤, 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 if-wif 1006 ⊤wtru 1476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-tru 1478 |
This theorem is referenced by: ifpdfbi 36837 |
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