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Metamath Proof Explorer

Theorem iuneq12d

Description: Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015) Remove DV conditions (Revised by GG, 1-Sep-2025)

		Ref	Expression
	Hypotheses	iuneq12d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
		iuneq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
	Assertion	iuneq12d	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		iuneq12d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		iuneq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3	1	eleq2d	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )
4	3	anbi1d	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐶 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑡 ∈ 𝐶 ) ) )
5	4	rexbidv2	⊢ ( 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-iun	⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = { 𝑡 ∣ ∃ 𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 }
8		df-iun	⊢ ∪ 𝑥 ∈ 𝐵 𝐶 = { 𝑡 ∣ ∃ 𝑥 ∈ 𝐵 𝑡 ∈ 𝐶 }
9	6 7 8	3eqtr4g	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶 )
10	2	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝐶 = 𝐷 )
11	10	iuneq2dv	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐵 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷 )
12	9 11	eqtrd	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷 )