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Mirrors > Home > MPE Home > Th. List > dfifp3 | Structured version Visualization version GIF version |
Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.) |
Ref | Expression |
---|---|
dfifp3 | ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfifp2 1008 | . 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))) | |
2 | pm4.64 386 | . . 3 ⊢ ((¬ 𝜑 → 𝜒) ↔ (𝜑 ∨ 𝜒)) | |
3 | 2 | anbi2i 726 | . 2 ⊢ (((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒))) |
4 | 1, 3 | bitri 263 | 1 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 if-wif 1006 |
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 196 df-or 384 df-an 385 df-ifp 1007 |
This theorem is referenced by: dfifp4 1010 ifptru 1017 |
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