Theorem caov31

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	caov31	⊢ ( ( 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 3 2 5	caovass	⊢ ( ( 𝐴 𝐹 𝐶 ) 𝐹 𝐵 ) = ( 𝐴 𝐹 ( 𝐶 𝐹 𝐵 ) )
7	1 3 2 4 5	caov12	⊢ ( 𝐴 𝐹 ( 𝐶 𝐹 𝐵 ) ) = ( 𝐶 𝐹 ( 𝐴 𝐹 𝐵 ) )
8	6 7	eqtri	⊢ ( ( 𝐴 𝐹 𝐶 ) 𝐹 𝐵 ) = ( 𝐶 𝐹 ( 𝐴 𝐹 𝐵 ) )
9	1 2 3 4 5	caov32	⊢ ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝐶 ) = ( ( 𝐴 𝐹 𝐶 ) 𝐹 𝐵 )
10	3 1 2 4 5	caov32	⊢ ( ( 𝐶 𝐹 𝐴 ) 𝐹 𝐵 ) = ( ( 𝐶 𝐹 𝐵 ) 𝐹 𝐴 )
11	3 1 2 5	caovass	⊢ ( ( 𝐶 𝐹 𝐴 ) 𝐹 𝐵 ) = ( 𝐶 𝐹 ( 𝐴 𝐹 𝐵 ) )
12	10 11	eqtr3i	⊢ ( ( 𝐶 𝐹 𝐵 ) 𝐹 𝐴 ) = ( 𝐶 𝐹 ( 𝐴 𝐹 𝐵 ) )
13	8 9 12	3eqtr4i	⊢ ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝐶 ) = ( ( 𝐶 𝐹 𝐵 ) 𝐹 𝐴 )