Theorem iuneq12i 36368

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 2827	. . . 4 ⊢ (𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷)
4	1, 3	rexeqbii 3314	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑡 ∈ 𝐷)
5	4	abbii 2802	. 2 ⊢ {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} = {𝑡 ∣ ∃𝑥 ∈ 𝐵 𝑡 ∈ 𝐷}
6		df-iun 4947	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶}
7		df-iun 4947	. 2 ⊢ ∪ 𝑥 ∈ 𝐵 𝐷 = {𝑡 ∣ ∃𝑥 ∈ 𝐵 𝑡 ∈ 𝐷}
8	5, 6, 7	3eqtr4i 2768	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷

Syntax hints:   = wceq 1542   ∈ wcel 2114  {cab 2713  ∃wrex 3059  ∪ ciun 4945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-rex 3060  df-iun 4947

This theorem is referenced by: (None)