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Mirrors > Home > MPE Home > Th. List > anifp | Structured version Visualization version GIF version |
Description: The conditional operator is implied by the conjunction of its possible outputs. Dual statement of ifpor 1070. (Contributed by BJ, 30-Sep-2019.) |
Ref | Expression |
---|---|
anifp | ⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | olc 865 | . . 3 ⊢ (𝜓 → (¬ 𝜑 ∨ 𝜓)) | |
2 | olc 865 | . . 3 ⊢ (𝜒 → (𝜑 ∨ 𝜒)) | |
3 | 1, 2 | anim12i 613 | . 2 ⊢ ((𝜓 ∧ 𝜒) → ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒))) |
4 | dfifp4 1064 | . 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒))) | |
5 | 3, 4 | sylibr 233 | 1 ⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 844 if-wif 1060 |
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 206 df-an 397 df-or 845 df-ifp 1061 |
This theorem is referenced by: bj-consensus 34747 bj-consensusALT 34748 axfrege58a 41444 |
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