Theorem iuneq12i 37199

Description: Equality theorem for indexed union. Inference version. (Contributed by GG, 1-Sep-2025.)

Hypotheses

Ref	Expression
iuneq12i.1	⊢ 𝐴 = 𝐵
iuneq12i.2	⊢ 𝐶 = 𝐷

Assertion

Ref	Expression
iuneq12i	⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷

Proof of Theorem iuneq12i

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iuneq12i.1	. . . 4 ⊢ 𝐴 = 𝐵
2		iuneq12i.2	. . . . 5 ⊢ 𝐶 = 𝐷
3	2	eleq2i 2856	. . . 4 ⊢ (𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷)
4	1, 3	rexeqbii 3337	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑡 ∈ 𝐷)
5	4	abbii 2830	. 2 ⊢ {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} = {𝑡 ∣ ∃𝑥 ∈ 𝐵 𝑡 ∈ 𝐷}
6		df-iun 4958	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶}
7		df-iun 4958	. 2 ⊢ ∪ 𝑥 ∈ 𝐵 𝐷 = {𝑡 ∣ ∃𝑥 ∈ 𝐵 𝑡 ∈ 𝐷}
8	5, 6, 7	3eqtr4i 2796	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  {cab 2741  ∃wrex 3090  ∪ ciun 4956

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rex 3091  df-iun 4958

This theorem is used by: (None)