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Metamath Proof Explorer
Description: Analogue of an4 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011) (Proof shortened by Andrew Salmon, 25-May-2011)
|
|
Ref |
Expression |
|
Assertion |
3an6 |
⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜒 ∧ 𝜃 ) ∧ ( 𝜏 ∧ 𝜂 ) ) ↔ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) ∧ ( 𝜓 ∧ 𝜃 ∧ 𝜂 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
an6 |
⊢ ( ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) ∧ ( 𝜓 ∧ 𝜃 ∧ 𝜂 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜒 ∧ 𝜃 ) ∧ ( 𝜏 ∧ 𝜂 ) ) ) |
| 2 |
1
|
bicomi |
⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜒 ∧ 𝜃 ) ∧ ( 𝜏 ∧ 𝜂 ) ) ↔ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) ∧ ( 𝜓 ∧ 𝜃 ∧ 𝜂 ) ) ) |