Theorem iuneq12f

Description: Equality deduction for indexed unions. (Contributed by Giovanni Mascellani, 10-Apr-2018)

	Ref	Expression
Hypotheses	iuneq12f.1	⊢ Ⅎ 𝑥 𝐴
	iuneq12f.2	⊢ Ⅎ 𝑥 𝐵
Assertion	iuneq12f	⊢ ( ( 𝐴 = 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 𝐶 = 𝐷 ) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		iuneq12f.1	⊢ Ⅎ 𝑥 𝐴
2		iuneq12f.2	⊢ Ⅎ 𝑥 𝐵
3		iuneq2	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐶 = 𝐷 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐴 𝐷 )
4	1 2	iuneq2f	⊢ ( 𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐷 = ∪ 𝑥 ∈ 𝐵 𝐷 )
5	3 4	sylan9eqr	⊢ ( ( 𝐴 = 𝐵 ∧ ∀ 𝑥 ∈ 𝐴 𝐶 = 𝐷 ) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷 )