Theorem iuneq2i

Description: Equality inference for indexed union. (Contributed by NM, 22-Oct-2003)

	Ref	Expression
Hypothesis	iuneq2i.1	⊢ ( 𝑥 ∈ 𝐴 → 𝐵 = 𝐶 )
Assertion	iuneq2i	⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶

Proof

Step	Hyp	Ref	Expression
1		iuneq2i.1	⊢ ( 𝑥 ∈ 𝐴 → 𝐵 = 𝐶 )
2		iuneq2	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶 )
3	2 1	mprg	⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶