Metamath Proof Explorer

Theorem caov411d

Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995) (Revised by Mario Carneiro, 30-Dec-2014)

		Ref	Expression
	Hypotheses	caovd.1	$⊢ φ \to A \in S$
		caovd.2	$⊢ φ \to B \in S$
		caovd.3	$⊢ φ \to C \in S$
		caovd.com	$⊢ (φ \land (x \in S \land y \in S)) \to x F y = y F x$
		caovd.ass	$⊢ (φ \land (x \in S \land y \in S \land z \in S)) \to (x F y) F z = x F (y F z)$
		caovd.4	$⊢ φ \to D \in S$
		caovd.cl	$⊢ (φ \land (x \in S \land y \in S)) \to x F y \in S$
	Assertion	caov411d	$⊢ φ \to (A F B) F (C F D) = (C F B) F (A F D)$

Proof

Step	Hyp	Ref	Expression
1		caovd.1	$⊢ φ \to A \in S$
2		caovd.2	$⊢ φ \to B \in S$
3		caovd.3	$⊢ φ \to C \in S$
4		caovd.com	$⊢ (φ \land (x \in S \land y \in S)) \to x F y = y F x$
5		caovd.ass	$⊢ (φ \land (x \in S \land y \in S \land z \in S)) \to (x F y) F z = x F (y F z)$
6		caovd.4	$⊢ φ \to D \in S$
7		caovd.cl	$⊢ (φ \land (x \in S \land y \in S)) \to x F y \in S$
8	2 1 3 4 5 6 7	caov4d	$⊢ φ \to (B F A) F (C F D) = (B F C) F (A F D)$
9	4 2 1	caovcomd	$⊢ φ \to B F A = A F B$
10	9	oveq1d	$⊢ φ \to (B F A) F (C F D) = (A F B) F (C F D)$
11	4 2 3	caovcomd	$⊢ φ \to B F C = C F B$
12	11	oveq1d	$⊢ φ \to (B F C) F (A F D) = (C F B) F (A F D)$
13	8 10 12	3eqtr3d	$⊢ φ \to (A F B) F (C F D) = (C F B) F (A F D)$