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 ) )