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Mirrors > Home > MPE Home > Th. List > Mathboxes > ifpan123g | Structured version Visualization version GIF version |
Description: Conjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.) |
Ref | Expression |
---|---|
ifpan123g | ⊢ ((if-(𝜑, 𝜒, 𝜏) ∧ if-(𝜓, 𝜃, 𝜂)) ↔ (((¬ 𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜏)) ∧ ((¬ 𝜓 ∨ 𝜃) ∧ (𝜓 ∨ 𝜂)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfifp4 1066 | . 2 ⊢ (if-(𝜑, 𝜒, 𝜏) ↔ ((¬ 𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜏))) | |
2 | dfifp4 1066 | . 2 ⊢ (if-(𝜓, 𝜃, 𝜂) ↔ ((¬ 𝜓 ∨ 𝜃) ∧ (𝜓 ∨ 𝜂))) | |
3 | 1, 2 | anbi12i 628 | 1 ⊢ ((if-(𝜑, 𝜒, 𝜏) ∧ if-(𝜓, 𝜃, 𝜂)) ↔ (((¬ 𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜏)) ∧ ((¬ 𝜓 ∨ 𝜃) ∧ (𝜓 ∨ 𝜂)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 397 ∨ wo 846 if-wif 1062 |
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 398 df-or 847 df-ifp 1063 |
This theorem is referenced by: ifpan23 41806 |
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