Theorem cbviunvw2 37114

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 2846	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷))
4	1, 3	cbvrexvw2 37109	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐵 𝑡 ∈ 𝐷)
5	4	abbii 2827	. 2 ⊢ {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶} = {𝑡 ∣ ∃𝑦 ∈ 𝐵 𝑡 ∈ 𝐷}
6		df-iun 4952	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐶}
7		df-iun 4952	. 2 ⊢ ∪ 𝑦 ∈ 𝐵 𝐷 = {𝑡 ∣ ∃𝑦 ∈ 𝐵 𝑡 ∈ 𝐷}
8	5, 6, 7	3eqtr4i 2793	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑦 ∈ 𝐵 𝐷

This proof depends on syntax axioms:   → wi 4   = 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: (None)