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Metamath Proof Explorer

Theorem caov32

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 caov32 ⊢ A F B F C = A F C 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 2 3 4 caovcom ⊢ B F C = C F B
7 6 oveq2i ⊢ A F B F C = A F C F B
8 1 2 3 5 caovass ⊢ A F B F C = A F B F C
9 1 3 2 5 caovass ⊢ A F C F B = A F C F B
10 7 8 9 3eqtr4i ⊢ A F B F C = A F C F B