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Mirrors > Home > MPE Home > Th. List > dfifp5 | Structured version Visualization version GIF version |
Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.) |
Ref | Expression |
---|---|
dfifp5 | ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (¬ 𝜑 → 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfifp2 1045 | . 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒))) | |
2 | imor 839 | . . 3 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓)) | |
3 | 2 | anbi1i 614 | . 2 ⊢ (((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (¬ 𝜑 → 𝜒))) |
4 | 1, 3 | bitri 267 | 1 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (¬ 𝜑 → 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 ∨ wo 833 if-wif 1043 |
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 199 df-an 388 df-or 834 df-ifp 1044 |
This theorem is referenced by: (None) |
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