Theorem an42 587

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 586	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)))
2		ancom 266	. . 3 ⊢ ((𝜓 ∧ 𝜃) ↔ (𝜃 ∧ 𝜓))
3	2	anbi2i 457	. 2 ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓)))
4	1, 3	bitri 184	1 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓)))

Syntax hints:   ∧ wa 104   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  rnlem  976  supmoti  6994  distrnqg  7388  distrnq0  7460  prcdnql  7485  prcunqu  7486