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Mirrors > Home > MPE Home > Th. List > Mathboxes > 3an4ancom24 | Structured version Visualization version GIF version |
Description: Commutative law for a conjunction with a triple conjunction: second and forth positions interchanged. (Contributed by AV, 18-Feb-2022.) |
Ref | Expression |
---|---|
3an4ancom24 | ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ↔ ((𝜑 ∧ 𝜃 ∧ 𝜒) ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | an4com24 44728 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜃) ∧ (𝜒 ∧ 𝜓))) | |
2 | 3an4anass 1104 | . 2 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃))) | |
3 | 3an4anass 1104 | . 2 ⊢ (((𝜑 ∧ 𝜃 ∧ 𝜒) ∧ 𝜓) ↔ ((𝜑 ∧ 𝜃) ∧ (𝜒 ∧ 𝜓))) | |
4 | 1, 2, 3 | 3bitr4i 303 | 1 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ↔ ((𝜑 ∧ 𝜃 ∧ 𝜒) ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∧ w3a 1086 |
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-3an 1088 |
This theorem is referenced by: (None) |
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