Theorem caov12

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	caov12	⊢ ( 𝐴 𝐹 ( 𝐵 𝐹 𝐶 ) ) = ( 𝐵 𝐹 ( 𝐴 𝐹 𝐶 ) )

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 4	caovcom	⊢ ( 𝐴 𝐹 𝐵 ) = ( 𝐵 𝐹 𝐴 )
7	6	oveq1i	⊢ ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝐶 ) = ( ( 𝐵 𝐹 𝐴 ) 𝐹 𝐶 )
8	1 2 3 5	caovass	⊢ ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝐶 ) = ( 𝐴 𝐹 ( 𝐵 𝐹 𝐶 ) )
9	2 1 3 5	caovass	⊢ ( ( 𝐵 𝐹 𝐴 ) 𝐹 𝐶 ) = ( 𝐵 𝐹 ( 𝐴 𝐹 𝐶 ) )
10	7 8 9	3eqtr3i	⊢ ( 𝐴 𝐹 ( 𝐵 𝐹 𝐶 ) ) = ( 𝐵 𝐹 ( 𝐴 𝐹 𝐶 ) )