Theorem caov4

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	caov4	⊢ ( ( 𝐴 𝐹 𝐵 ) 𝐹 ( 𝐶 𝐹 𝐷 ) ) = ( ( 𝐴 𝐹 𝐶 ) 𝐹 ( 𝐵 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	2 3 6 4 5	caov12	⊢ ( 𝐵 𝐹 ( 𝐶 𝐹 𝐷 ) ) = ( 𝐶 𝐹 ( 𝐵 𝐹 𝐷 ) )
8	7	oveq2i	⊢ ( 𝐴 𝐹 ( 𝐵 𝐹 ( 𝐶 𝐹 𝐷 ) ) ) = ( 𝐴 𝐹 ( 𝐶 𝐹 ( 𝐵 𝐹 𝐷 ) ) )
9		ovex	⊢ ( 𝐶 𝐹 𝐷 ) ∈ V
10	1 2 9 5	caovass	⊢ ( ( 𝐴 𝐹 𝐵 ) 𝐹 ( 𝐶 𝐹 𝐷 ) ) = ( 𝐴 𝐹 ( 𝐵 𝐹 ( 𝐶 𝐹 𝐷 ) ) )
11		ovex	⊢ ( 𝐵 𝐹 𝐷 ) ∈ V
12	1 3 11 5	caovass	⊢ ( ( 𝐴 𝐹 𝐶 ) 𝐹 ( 𝐵 𝐹 𝐷 ) ) = ( 𝐴 𝐹 ( 𝐶 𝐹 ( 𝐵 𝐹 𝐷 ) ) )
13	8 10 12	3eqtr4i	⊢ ( ( 𝐴 𝐹 𝐵 ) 𝐹 ( 𝐶 𝐹 𝐷 ) ) = ( ( 𝐴 𝐹 𝐶 ) 𝐹 ( 𝐵 𝐹 𝐷 ) )