Metamath Proof Explorer

Theorem iineq12dv

Description: Equality deduction for indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021) Remove DV conditions. (Revised by GG, 1-Sep-2025)

		Ref	Expression
	Hypotheses	iineq12dv.1	$⊢ φ \to A = B$
		iineq12dv.2	$⊢ (φ \land x \in B) \to C = D$
	Assertion	iineq12dv	$⊢ φ \to ⋂_{x \in A} C = ⋂_{x \in B} D$

Proof

Step	Hyp	Ref	Expression
1		iineq12dv.1	$⊢ φ \to A = B$
2		iineq12dv.2	$⊢ (φ \land x \in B) \to C = D$
3	1	eleq2d	$⊢ φ \to (x \in A \leftrightarrow x \in B)$
4	3	imbi1d	$⊢ φ \to ((x \in A \to t \in C) \leftrightarrow (x \in B \to t \in C))$
5	4	ralbidv2	$⊢ φ \to (\forall x \in A t \in C \leftrightarrow \forall x \in B t \in C)$
6	5	abbidv	$⊢ φ \to \{t \| \forall x \in A t \in C\} = \{t \| \forall x \in B t \in C\}$
7		df-iin	$⊢ ⋂_{x \in A} C = \{t \| \forall x \in A t \in C\}$
8		df-iin	$⊢ ⋂_{x \in B} C = \{t \| \forall x \in B t \in C\}$
9	6 7 8	3eqtr4g	$⊢ φ \to ⋂_{x \in A} C = ⋂_{x \in B} C$
10	2	iineq2dv	$⊢ φ \to ⋂_{x \in B} C = ⋂_{x \in B} D$
11	9 10	eqtrd	$⊢ φ \to ⋂_{x \in A} C = ⋂_{x \in B} D$