Theorem cbviunvw2 36265

Description: Change bound variable and domain in indexed unions, using implicit substitution. (Contributed by GG, 14-Aug-2025.)

Hypotheses

Ref	Expression
cbviunvw2.1	⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)
cbviunvw2.2	⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)

Assertion

Ref	Expression
cbviunvw2	⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑦 ∈ 𝐵 𝐷

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑥,𝐵   𝑦,𝐶   𝑥,𝐷

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem cbviunvw2

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbviunvw2.2	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
2		cbviunvw2.1	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)
3	2	eleq2d 2817	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷))
4	1, 3	cbvrexvw2 36260	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐵 𝑡 ∈ 𝐷)
5	4	abbii 2798	. 2 ⊢ {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} = {𝑡 ∣ ∃𝑦 ∈ 𝐵 𝑡 ∈ 𝐷}
6		df-iun 4943	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶}
7		df-iun 4943	. 2 ⊢ ∪ 𝑦 ∈ 𝐵 𝐷 = {𝑡 ∣ ∃𝑦 ∈ 𝐵 𝑡 ∈ 𝐷}
8	5, 6, 7	3eqtr4i 2764	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑦 ∈ 𝐵 𝐷

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  {cab 2709  ∃wrex 3056  ∪ ciun 4941

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rex 3057  df-iun 4943

This theorem is referenced by: (None)