Theorem iuneq2df

Description: Equality deduction for indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	iuneq2df.1	⊢ Ⅎ 𝑥 𝜑
	iuneq2df.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐶 )
Assertion	iuneq2df	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		iuneq2df.1	⊢ Ⅎ 𝑥 𝜑
2		iuneq2df.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐶 )
3	2	ex	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → 𝐵 = 𝐶 ) )
4	1 3	ralrimi	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 )
5		iuneq2	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶 )
6	4 5	syl	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐶 )