Theorem cbviun 3903

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.)

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 groups:   𝑦,𝐴   𝑥,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 2302	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
3		cbviun.2	. . . . 5 ⊢ Ⅎ𝑥𝐶
4	3	nfcri 2302	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶
5		cbviun.3	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
6	5	eleq2d 2236	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶))
7	2, 4, 6	cbvrex 2689	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶)
8	7	abbii 2282	. 2 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶}
9		df-iun 3868	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
10		df-iun 3868	. 2 ⊢ ∪ 𝑦 ∈ 𝐴 𝐶 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶}
11	8, 9, 10	3eqtr4i 2196	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   = wceq 1343   ∈ wcel 2136  {cab 2151  Ⅎwnfc 2295  ∃wrex 2445  ∪ ciun 3866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-iun 3868

This theorem is referenced by:  cbviunv  3905  funiunfvdmf  5732  mpomptsx  6165  dmmpossx  6167  fmpox  6168  fsum2dlemstep  11375  fisumcom2  11379  fsumiun  11418  fprod2dlemstep  11563  fprodcom2fi  11567  ctiunctlemf  12371  ctiunctal  12374