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Mirrors > Home > MPE Home > Th. List > Mathboxes > ifpdfor | Structured version Visualization version GIF version |
Description: Define or in terms of conditional logic operator and true. (Contributed by RP, 20-Apr-2020.) |
Ref | Expression |
---|---|
ifpdfor | ⊢ ((𝜑 ∨ 𝜓) ↔ if-(𝜑, ⊤, 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tru 1546 | . . . 4 ⊢ ⊤ | |
2 | 1 | olci 865 | . . 3 ⊢ (¬ 𝜑 ∨ ⊤) |
3 | 2 | biantrur 534 | . 2 ⊢ ((𝜑 ∨ 𝜓) ↔ ((¬ 𝜑 ∨ ⊤) ∧ (𝜑 ∨ 𝜓))) |
4 | dfifp4 1066 | . 2 ⊢ (if-(𝜑, ⊤, 𝜓) ↔ ((¬ 𝜑 ∨ ⊤) ∧ (𝜑 ∨ 𝜓))) | |
5 | 3, 4 | bitr4i 281 | 1 ⊢ ((𝜑 ∨ 𝜓) ↔ if-(𝜑, ⊤, 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∧ wa 399 ∨ wo 846 if-wif 1062 ⊤wtru 1543 |
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 847 df-ifp 1063 df-tru 1545 |
This theorem is referenced by: ifpdfxor 40632 |
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