Theorem cbviun 4996

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 2404. See cbviung 4998 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 2918	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
3		cbviun.2	. . . . 5 ⊢ Ⅎ𝑥𝐶
4	3	nfcri 2918	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶
5		cbviun.3	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
6	5	eleq2d 2850	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶))
7	2, 4, 6	cbvrexw 3309	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶)
8	7	abbii 2830	. 2 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶}
9		df-iun 4956	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
10		df-iun 4956	. 2 ⊢ ∪ 𝑦 ∈ 𝐴 𝐶 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶}
11	8, 9, 10	3eqtr4i 2796	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  {cab 2741  Ⅎwnfc 2911  ∃wrex 3090  ∪ ciun 4954

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-iun 4956

This theorem is used by:  disjxiun  5103  funiunfvf  7276  mpomptsx  8100  dmmpossx  8102  fmpox  8103  ovmptssOLD  8130  iunfi  9360  fsum2dlem  15965  fsumcom2  15969  fsumiun  16017  fprod2dlem  16176  fprodcom2  16180  gsumcom2  20219  fiuncmp  23752  ovolfiniun  25852  ovoliunlem3  25855  ovoliun  25856  finiunmbl  25895  volfiniun  25898  iunmbl  25904  limciun  26244  iunxpssiun1  33234  iuneqfzuzlem  46551  fsumiunss  46792  sge0iunmpt  47633  meaiunincf  47698  meaiuninc3  47700  smfliminf  48046  dmmpossx2  49654