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Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-ifp4impr | Structured version Visualization version GIF version |
Description: If one case of an if- condition is a consequence of the other, the expression in dfifp4 1063 can be shortened. (Contributed by Wolf Lammen, 18-Jun-2024.) |
Ref | Expression |
---|---|
wl-ifp4impr | ⊢ ((𝜒 → 𝜓) → (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∨ 𝜒) ∧ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wl-ifpimpr 35148 | . 2 ⊢ ((𝜒 → 𝜓) → (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ 𝜒))) | |
2 | pm4.71 562 | . . . . 5 ⊢ ((𝜒 → 𝜓) ↔ (𝜒 ↔ (𝜒 ∧ 𝜓))) | |
3 | 2 | biimpi 219 | . . . 4 ⊢ ((𝜒 → 𝜓) → (𝜒 ↔ (𝜒 ∧ 𝜓))) |
4 | 3 | orbi2d 914 | . . 3 ⊢ ((𝜒 → 𝜓) → (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ 𝜓)))) |
5 | andir 1007 | . . 3 ⊢ (((𝜑 ∨ 𝜒) ∧ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ 𝜓))) | |
6 | 4, 5 | bitr4di 293 | . 2 ⊢ ((𝜒 → 𝜓) → (((𝜑 ∧ 𝜓) ∨ 𝜒) ↔ ((𝜑 ∨ 𝜒) ∧ 𝜓))) |
7 | 1, 6 | bitrd 282 | 1 ⊢ ((𝜒 → 𝜓) → (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∨ 𝜒) ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 845 if-wif 1059 |
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 210 df-an 401 df-or 846 df-ifp 1060 |
This theorem is referenced by: wl-df3-3mintru2 35168 |
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