![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ifpid2 | Structured version Visualization version GIF version |
Description: Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
Ref | Expression |
---|---|
ifpid2 | ⊢ (𝜑 ↔ if-(𝜑, ⊤, ⊥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tru 1546 | . . . 4 ⊢ ⊤ | |
2 | 1 | olci 865 | . . 3 ⊢ (¬ 𝜑 ∨ ⊤) |
3 | 2 | biantrur 532 | . 2 ⊢ ((𝜑 ∨ ⊥) ↔ ((¬ 𝜑 ∨ ⊤) ∧ (𝜑 ∨ ⊥))) |
4 | fal 1556 | . . 3 ⊢ ¬ ⊥ | |
5 | 4 | biorfi 938 | . 2 ⊢ (𝜑 ↔ (𝜑 ∨ ⊥)) |
6 | dfifp4 1066 | . 2 ⊢ (if-(𝜑, ⊤, ⊥) ↔ ((¬ 𝜑 ∨ ⊤) ∧ (𝜑 ∨ ⊥))) | |
7 | 3, 5, 6 | 3bitr4i 303 | 1 ⊢ (𝜑 ↔ if-(𝜑, ⊤, ⊥)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 397 ∨ wo 846 if-wif 1062 ⊤wtru 1543 ⊥wfal 1554 |
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 206 df-an 398 df-or 847 df-ifp 1063 df-tru 1545 df-fal 1555 |
This theorem is referenced by: frege52aid 42204 |
Copyright terms: Public domain | W3C validator |