Theorem cbviun 5012

Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.) Add disjoint variable condition to avoid ax-13 2376. See cbviung 5014 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.)

Hypotheses

Ref	Expression
cbviun.1	⊢ Ⅎ𝑦𝐵
cbviun.2	⊢ Ⅎ𝑥𝐶
cbviun.3	⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)

Assertion

Ref	Expression
cbviun	⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶

Distinct variable group:   𝑥,𝑦,𝐴

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

Proof of Theorem cbviun

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbviun.1	. . . . 5 ⊢ Ⅎ𝑦𝐵
2	1	nfcri 2890	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
3		cbviun.2	. . . . 5 ⊢ Ⅎ𝑥𝐶
4	3	nfcri 2890	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶
5		cbviun.3	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
6	5	eleq2d 2820	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶))
7	2, 4, 6	cbvrexw 3287	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶)
8	7	abbii 2802	. 2 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶}
9		df-iun 4969	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
10		df-iun 4969	. 2 ⊢ ∪ 𝑦 ∈ 𝐴 𝐶 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶}
11	8, 9, 10	3eqtr4i 2768	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  {cab 2713  Ⅎwnfc 2883  ∃wrex 3060  ∪ ciun 4967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-12 2177  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-iun 4969

This theorem is referenced by:  disjxiun  5116  funiunfvf  7241  mpomptsx  8063  dmmpossx  8065  fmpox  8066  ovmptss  8092  iunfi  9355  fsum2dlem  15786  fsumcom2  15790  fsumiun  15837  fprod2dlem  15996  fprodcom2  16000  gsumcom2  19956  fiuncmp  23342  ovolfiniun  25454  ovoliunlem3  25457  ovoliun  25458  finiunmbl  25497  volfiniun  25500  iunmbl  25506  limciun  25847  iunxpssiun1  32549  iuneqfzuzlem  45361  fsumiunss  45604  sge0iunmpt  46447  meaiunincf  46512  meaiuninc3  46514  smfliminf  46860  dmmpossx2  48312