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Mirrors > Home > MPE Home > Th. List > dfifp4 | Structured version Visualization version GIF version |
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 1060 | . 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒))) | |
2 | imor 849 | . . 3 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓)) | |
3 | 2 | anbi1i 625 | . 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜑 ∨ 𝜒)) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒))) |
4 | 1, 3 | bitri 277 | 1 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 if-wif 1057 |
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 209 df-an 399 df-or 844 df-ifp 1058 |
This theorem is referenced by: anifp 1065 ifpan123g 39817 ifpan23 39818 ifpdfor2 39819 ifpdfor 39823 ifpim1 39827 ifpnot 39828 ifpid2 39829 ifpim2 39830 ifpnot23 39837 ifpidg 39850 ifpim123g 39859 ifpimim 39868 |
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