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Mirrors > Home > MPE Home > Th. List > Mathboxes > ifpdfor2 | Structured version Visualization version GIF version |
Description: Define or in terms of conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
Ref | Expression |
---|---|
ifpdfor2 | ⊢ ((𝜑 ∨ 𝜓) ↔ if-(𝜑, 𝜑, 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.1 896 | . . 3 ⊢ (¬ 𝜑 ∨ 𝜑) | |
2 | 1 | biantrur 532 | . 2 ⊢ ((𝜑 ∨ 𝜓) ↔ ((¬ 𝜑 ∨ 𝜑) ∧ (𝜑 ∨ 𝜓))) |
3 | dfifp4 1066 | . 2 ⊢ (if-(𝜑, 𝜑, 𝜓) ↔ ((¬ 𝜑 ∨ 𝜑) ∧ (𝜑 ∨ 𝜓))) | |
4 | 2, 3 | bitr4i 278 | 1 ⊢ ((𝜑 ∨ 𝜓) ↔ if-(𝜑, 𝜑, 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 397 ∨ wo 846 if-wif 1062 |
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 |
This theorem is referenced by: ifporcor 41808 |
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