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Mirrors > Home > MPE Home > Th. List > dfifp6 | Structured version Visualization version GIF version |
Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.) |
Ref | Expression |
---|---|
dfifp6 | ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜒 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ifp 1013 | . 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒))) | |
2 | ancom 466 | . . . 4 ⊢ ((¬ 𝜑 ∧ 𝜒) ↔ (𝜒 ∧ ¬ 𝜑)) | |
3 | annim 441 | . . . 4 ⊢ ((𝜒 ∧ ¬ 𝜑) ↔ ¬ (𝜒 → 𝜑)) | |
4 | 2, 3 | bitri 264 | . . 3 ⊢ ((¬ 𝜑 ∧ 𝜒) ↔ ¬ (𝜒 → 𝜑)) |
5 | 4 | orbi2i 541 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜒 → 𝜑))) |
6 | 1, 5 | bitri 264 | 1 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜒 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 if-wif 1012 |
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 197 df-or 385 df-an 386 df-ifp 1013 |
This theorem is referenced by: dfifp7 1019 ifpdfan2 37807 |
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