Metamath Proof Explorer


Theorem an4

Description: Rearrangement of 4 conjuncts. (Contributed by NM, 10-Jul-1994)

Ref Expression
Assertion an4 ( ( ( 𝜑𝜓 ) ∧ ( 𝜒𝜃 ) ) ↔ ( ( 𝜑𝜒 ) ∧ ( 𝜓𝜃 ) ) )

Proof

Step Hyp Ref Expression
1 anass ( ( ( 𝜑𝜓 ) ∧ ( 𝜒𝜃 ) ) ↔ ( 𝜑 ∧ ( 𝜓 ∧ ( 𝜒𝜃 ) ) ) )
2 an12 ( ( 𝜓 ∧ ( 𝜒𝜃 ) ) ↔ ( 𝜒 ∧ ( 𝜓𝜃 ) ) )
3 2 bianass ( ( 𝜑 ∧ ( 𝜓 ∧ ( 𝜒𝜃 ) ) ) ↔ ( ( 𝜑𝜒 ) ∧ ( 𝜓𝜃 ) ) )
4 1 3 bitri ( ( ( 𝜑𝜓 ) ∧ ( 𝜒𝜃 ) ) ↔ ( ( 𝜑𝜒 ) ∧ ( 𝜓𝜃 ) ) )