Metamath Proof Explorer

Theorem caovassd

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

	Ref	Expression
Hypotheses	caovassg.1	$⊢ (φ \land (x \in S \land y \in S \land z \in S)) \to (x F y) F z = x F (y F z)$
	caovassd.2	$⊢ φ \to A \in S$
	caovassd.3	$⊢ φ \to B \in S$
	caovassd.4	$⊢ φ \to C \in S$
Assertion	caovassd	$⊢ φ \to (A F B) F C = A F (B F C)$

Step	Hyp	Ref	Expression
1		caovassg.1	$⊢ (φ \land (x \in S \land y \in S \land z \in S)) \to (x F y) F z = x F (y F z)$
2		caovassd.2	$⊢ φ \to A \in S$
3		caovassd.3	$⊢ φ \to B \in S$
4		caovassd.4	$⊢ φ \to C \in S$
5		id	$⊢ φ \to φ$
6	1	caovassg	$⊢ (φ \land (A \in S \land B \in S \land C \in S)) \to (A F B) F C = A F (B F C)$
7	5 2 3 4 6	syl13anc	$⊢ φ \to (A F B) F C = A F (B F C)$