Metamath Proof Explorer

Theorem caov42

Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995)

Ref Expression
Hypotheses caov.1 ⊢ A ∈ V
caov.2 ⊢ B ∈ V
caov.3 ⊢ C ∈ V
caov.com ⊢ x F y = y F x
caov.ass ⊢ x F y F z = x F y F z
caov.4 ⊢ D ∈ V
Assertion caov42 ⊢ A F B F C F D = A F C F D F B

Proof

Step Hyp Ref Expression
1 caov.1 ⊢ A ∈ V
2 caov.2 ⊢ B ∈ V
3 caov.3 ⊢ C ∈ V
4 caov.com ⊢ x F y = y F x
5 caov.ass ⊢ x F y F z = x F y F z
6 caov.4 ⊢ D ∈ V
7 1 2 3 4 5 6 caov4 ⊢ A F B F C F D = A F C F B F D
8 2 6 4 caovcom ⊢ B F D = D F B
9 8 oveq2i ⊢ A F C F B F D = A F C F D F B
10 7 9 eqtri ⊢ A F B F C F D = A F C F D F B