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Metamath Proof Explorer

Theorem caov13

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
Assertion caov13 ⊢ A F B F C = C F B F A

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 1 2 3 4 5 caov31 ⊢ A F B F C = C F B F A
7 1 2 3 5 caovass ⊢ A F B F C = A F B F C
8 3 2 1 5 caovass ⊢ C F B F A = C F B F A
9 6 7 8 3eqtr3i ⊢ A F B F C = C F B F A