Theorem cbviundavw 36627

Description: Change bound variable in indexed unions. Deduction form. (Contributed by GG, 14-Aug-2025.)

Hypothesis

Ref	Expression
cbviundavw.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)

Assertion

Ref	Expression
cbviundavw	⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶)

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

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

Proof of Theorem cbviundavw

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbviundavw.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)
2	1	eleq2d 2850	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑡 ∈ 𝐵 ↔ 𝑡 ∈ 𝐶))
3	2	cbvrexdva 3245	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑡 ∈ 𝐶))
4	3	abbidv 2830	. 2 ⊢ (𝜑 → {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐵} = {𝑡 ∣ ∃𝑦 ∈ 𝐴 𝑡 ∈ 𝐶})
5		df-iun 4953	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐵}
6		df-iun 4953	. 2 ⊢ ∪ 𝑦 ∈ 𝐴 𝐶 = {𝑡 ∣ ∃𝑦 ∈ 𝐴 𝑡 ∈ 𝐶}
7	4, 5, 6	3eqtr4g 2824	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  {cab 2742  ∃wrex 3088  ∪ ciun 4951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-iun 4953

This theorem is referenced by: (None)