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Mirrors > Home > MPE Home > Th. List > Mathboxes > ifpim2 | Structured version Visualization version GIF version |
Description: Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
Ref | Expression |
---|---|
ifpim2 | ⊢ ((𝜑 → 𝜓) ↔ if-(𝜓, ⊤, ¬ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tru 1547 | . . . 4 ⊢ ⊤ | |
2 | 1 | olci 866 | . . 3 ⊢ (¬ 𝜓 ∨ ⊤) |
3 | 2 | biantrur 534 | . 2 ⊢ ((𝜓 ∨ ¬ 𝜑) ↔ ((¬ 𝜓 ∨ ⊤) ∧ (𝜓 ∨ ¬ 𝜑))) |
4 | imor 853 | . . 3 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓)) | |
5 | orcom 870 | . . 3 ⊢ ((¬ 𝜑 ∨ 𝜓) ↔ (𝜓 ∨ ¬ 𝜑)) | |
6 | 4, 5 | bitri 278 | . 2 ⊢ ((𝜑 → 𝜓) ↔ (𝜓 ∨ ¬ 𝜑)) |
7 | dfifp4 1067 | . 2 ⊢ (if-(𝜓, ⊤, ¬ 𝜑) ↔ ((¬ 𝜓 ∨ ⊤) ∧ (𝜓 ∨ ¬ 𝜑))) | |
8 | 3, 6, 7 | 3bitr4i 306 | 1 ⊢ ((𝜑 → 𝜓) ↔ if-(𝜓, ⊤, ¬ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 847 if-wif 1063 ⊤wtru 1544 |
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 210 df-an 400 df-or 848 df-ifp 1064 df-tru 1546 |
This theorem is referenced by: (None) |
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