Theorem cbviundavw 36220

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 2830	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑡 ∈ 𝐵 ↔ 𝑡 ∈ 𝐶))
3	2	cbvrexdva 3246	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑡 ∈ 𝐶))
4	3	abbidv 2811	. 2 ⊢ (𝜑 → {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐵} = {𝑡 ∣ ∃𝑦 ∈ 𝐴 𝑡 ∈ 𝐶})
5		df-iun 5017	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑡 ∣ ∃𝑥 ∈ 𝐴 𝑡 ∈ 𝐵}
6		df-iun 5017	. 2 ⊢ ∪ 𝑦 ∈ 𝐴 𝐶 = {𝑡 ∣ ∃𝑦 ∈ 𝐴 𝑡 ∈ 𝐶}
7	4, 5, 6	3eqtr4g 2805	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  ∃wrex 3076  ∪ ciun 5015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-iun 5017

This theorem is referenced by: (None)