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| Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.) | 
| Ref | Expression | 
|---|---|
| dfifp4 | ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dfifp3 1066 | . 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒))) | |
| 2 | imor 854 | . 2 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓)) | |
| 3 | 1, 2 | bianbi 627 | 1 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 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 207 df-an 396 df-or 849 df-ifp 1064 | 
| This theorem is referenced by: anifp 1072 ifpan123g 43472 ifpan23 43473 ifpdfor2 43474 ifpdfor 43478 ifpim1 43482 ifpnot 43483 ifpid2 43484 ifpim2 43485 ifpnot23 43491 ifpidg 43504 ifpim123g 43513 ifpimim 43522 | 
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