Theorem an42 577

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 576	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)))
2		ancom 264	. . 3 ⊢ ((𝜓 ∧ 𝜃) ↔ (𝜃 ∧ 𝜓))
3	2	anbi2i 453	. 2 ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓)))
4	1, 3	bitri 183	1 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓)))

Syntax hints:   ∧ wa 103   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  rnlem  966  supmoti  6958  distrnqg  7328  distrnq0  7400  prcdnql  7425  prcunqu  7426