Metamath Proof Explorer

Theorem caovordd

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

	Ref	Expression
Hypotheses	caovordg.1	$⊢ (φ \land (x \in S \land y \in S \land z \in S)) \to (x R y \leftrightarrow (z F x) R (z F y))$
	caovordd.2	$⊢ φ \to A \in S$
	caovordd.3	$⊢ φ \to B \in S$
	caovordd.4	$⊢ φ \to C \in S$
Assertion	caovordd	$⊢ φ \to (A R B \leftrightarrow (C F A) R (C F B))$

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