Theorem iuneq1d

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

	Ref	Expression
Hypothesis	iuneq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	iuneq1d	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		iuneq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		iuneq1	⊢ ( 𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶 )
3	1 2	syl	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶 )