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Mirrors > Home > MPE Home > Th. List > Mathboxes > ifpid3g | Structured version Visualization version GIF version |
Description: Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.) |
Ref | Expression |
---|---|
ifpid3g | ⊢ ((𝜒 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ (((𝜑 ∧ 𝜓) → 𝜒) ∧ ((𝜑 ∧ 𝜒) → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | olc 868 | . . 3 ⊢ (𝜒 → (𝜑 ∨ 𝜒)) | |
2 | 1, 1 | pm3.2i 474 | . 2 ⊢ ((𝜒 → (𝜑 ∨ 𝜒)) ∧ (𝜒 → (𝜑 ∨ 𝜒))) |
3 | ifpidg 40783 | . 2 ⊢ ((𝜒 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ ((((𝜑 ∧ 𝜓) → 𝜒) ∧ ((𝜑 ∧ 𝜒) → 𝜓)) ∧ ((𝜒 → (𝜑 ∨ 𝜒)) ∧ (𝜒 → (𝜑 ∨ 𝜒))))) | |
4 | 2, 3 | mpbiran2 710 | 1 ⊢ ((𝜒 ↔ if-(𝜑, 𝜓, 𝜒)) ↔ (((𝜑 ∧ 𝜓) → 𝜒) ∧ ((𝜑 ∧ 𝜒) → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 847 if-wif 1063 |
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 |
This theorem is referenced by: (None) |
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