Theorem caovassd

Description: Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014)

	Ref	Expression
Hypotheses	caovassg.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( ( 𝑥 𝐹 𝑦 ) 𝐹 𝑧 ) = ( 𝑥 𝐹 ( 𝑦 𝐹 𝑧 ) ) )
	caovassd.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
	caovassd.3	⊢ ( 𝜑 → 𝐵 ∈ 𝑆 )
	caovassd.4	⊢ ( 𝜑 → 𝐶 ∈ 𝑆 )
Assertion	caovassd	⊢ ( 𝜑 → ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝐶 ) = ( 𝐴 𝐹 ( 𝐵 𝐹 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		caovassg.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( ( 𝑥 𝐹 𝑦 ) 𝐹 𝑧 ) = ( 𝑥 𝐹 ( 𝑦 𝐹 𝑧 ) ) )
2		caovassd.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
3		caovassd.3	⊢ ( 𝜑 → 𝐵 ∈ 𝑆 )
4		caovassd.4	⊢ ( 𝜑 → 𝐶 ∈ 𝑆 )
5		id	⊢ ( 𝜑 → 𝜑 )
6	1	caovassg	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) → ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝐶 ) = ( 𝐴 𝐹 ( 𝐵 𝐹 𝐶 ) ) )
7	5 2 3 4 6	syl13anc	⊢ ( 𝜑 → ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝐶 ) = ( 𝐴 𝐹 ( 𝐵 𝐹 𝐶 ) ) )