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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 → 𝐴 = 𝐵 )
		iineq12dv.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝐶 = 𝐷 )
	Assertion	iineq12dv	⊢ ( 𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		iineq12dv.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		iineq12dv.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝐶 = 𝐷 )
3	1	eleq2d	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )
4	3	imbi1d	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 → 𝑡 ∈ 𝐶 ) ↔ ( 𝑥 ∈ 𝐵 → 𝑡 ∈ 𝐶 ) ) )
5	4	ralbidv2	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∀ 𝑥 ∈ 𝐵 𝑡 ∈ 𝐶 ) )
6	5	abbidv	⊢ ( 𝜑 → { 𝑡 ∣ ∀ 𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 } = { 𝑡 ∣ ∀ 𝑥 ∈ 𝐵 𝑡 ∈ 𝐶 } )
7		df-iin	⊢ ∩ 𝑥 ∈ 𝐴 𝐶 = { 𝑡 ∣ ∀ 𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 }
8		df-iin	⊢ ∩ 𝑥 ∈ 𝐵 𝐶 = { 𝑡 ∣ ∀ 𝑥 ∈ 𝐵 𝑡 ∈ 𝐶 }
9	6 7 8	3eqtr4g	⊢ ( 𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶 )
10	2	iineq2dv	⊢ ( 𝜑 → ∩ 𝑥 ∈ 𝐵 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷 )
11	9 10	eqtrd	⊢ ( 𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷 )