Theorem iuneq1i 44239

Description: Equality theorem for indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypothesis

Ref	Expression
iuneq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
iuneq1i	⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iuneq1i

Step	Hyp	Ref	Expression
1		iuneq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		iuneq1 5013	. 2 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)
3	1, 2	ax-mp 5	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶

Syntax hints:   = wceq 1540  ∪ ciun 4997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rex 3070  df-v 3475  df-in 3955  df-ss 3965  df-iun 4999

This theorem is referenced by:  ovolval4lem1  45827