Theorem iuneq1i 46237

Description: Equality theorem for indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.) Remove DV conditions. (Revised by GG, 1-Sep-2025.)

Hypothesis

Ref	Expression
iuneq1i.1	⊢ 𝐴 = 𝐵

Assertion

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

Proof of Theorem iuneq1i

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iuneq1i.1	. . . . . 6 ⊢ 𝐴 = 𝐵
2	1	eleq2i 2852	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)
3	2	anbi1i 636	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑡 ∈ 𝐶))
4	3	rexbii2 3105	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑡 ∈ 𝐶)
5	4	abbii 2827	. 2 ⊢ {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} = {𝑡 ∣ ∃𝑥 ∈ 𝐵 𝑡 ∈ 𝐶}
6		df-iun 4952	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶}
7		df-iun 4952	. 2 ⊢ ∪ 𝑥 ∈ 𝐵 𝐶 = {𝑡 ∣ ∃𝑥 ∈ 𝐵 𝑡 ∈ 𝐶}
8	5, 6, 7	3eqtr4i 2793	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2144  {cab 2738  ∃wrex 3086  ∪ ciun 4950

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rex 3087  df-iun 4952

This theorem is used by:  ovolval4lem1  47795