Theorem iuneq2d

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

	Ref	Expression
Hypothesis	iuneq2d.2	⊢ ( 𝜑 → 𝐵 = 𝐶 )
Assertion	iuneq2d	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		iuneq2d.2	⊢ ( 𝜑 → 𝐵 = 𝐶 )
2	1	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐶 )
3	2	iuneq2dv	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶 )