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 ) )