Theorem iuneq12d

Description: Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015)

	Ref	Expression
Hypotheses	iuneq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	iuneq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	iuneq12d	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		iuneq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		iuneq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3	1	iuneq1d	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶 )
4	2	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝐶 = 𝐷 )
5	4	iuneq2dv	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐵 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷 )
6	3 5	eqtrd	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷 )