Theorem an42 667

Description: Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996.)

Assertion

Ref	Expression
an42	⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓)))

Proof of Theorem an42

Step	Hyp	Ref	Expression
1		an4 666	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)))
2		ancom 464	. . 3 ⊢ ((𝜓 ∧ 𝜃) ↔ (𝜃 ∧ 𝜓))
3	2	anbi2i 632	. 2 ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓)))
4	1, 3	bitri 277	1 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓)))

Syntax hints:   ↔ wb 208   ∧ wa 399

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 209  df-an 400

This theorem is referenced by:  an43  668  an33rean  1504  brecop2  8793  supmo  9398  infmo  9443  aceq1  10073  dfiso2  17805  eulerpartlemt0  34666  isbasisrelowllem1  37849  isbasisrelowllem2  37850  ifp1bi  44078