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Metamath Proof Explorer

Theorem prodeq2

Description: Equality theorem for product. (Contributed by Scott Fenton, 4-Dec-2017)

		Ref	Expression
	Assertion	prodeq2	$⊢ \forall k \in A B = C \to \prod_{k \in A} B = \prod_{k \in A} C$

Step	Hyp	Ref	Expression
1		fveq2	$⊢ B = C \to I (B) = I (C)$
2	1	ralimi	$⊢ \forall k \in A B = C \to \forall k \in A I (B) = I (C)$
3		prodeq2ii	$⊢ \forall k \in A I (B) = I (C) \to \prod_{k \in A} B = \prod_{k \in A} C$
4	2 3	syl	$⊢ \forall k \in A B = C \to \prod_{k \in A} B = \prod_{k \in A} C$