Metamath Proof Explorer

Theorem caov411

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 caov411 ⊢ A F B F C F D = C F B F A F D

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 caov31 ⊢ A F B F C = C F B F A
8 7 oveq1i ⊢ A F B F C F D = C F B F A F D
9 ovex ⊢ A F B ∈ V
10 9 3 6 5 caovass ⊢ A F B F C F D = A F B F C F D
11 ovex ⊢ C F B ∈ V
12 11 1 6 5 caovass ⊢ C F B F A F D = C F B F A F D
13 8 10 12 3eqtr3i ⊢ A F B F C F D = C F B F A F D