Theorem cbviunf 32541

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
cbviunf.x	⊢ Ⅎ𝑥𝐴
cbviunf.y	⊢ Ⅎ𝑦𝐴
cbviunf.1	⊢ Ⅎ𝑦𝐵
cbviunf.2	⊢ Ⅎ𝑥𝐶
cbviunf.3	⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)

Assertion

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

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem cbviunf

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbviunf.x	. . . 4 ⊢ Ⅎ𝑥𝐴
2		cbviunf.y	. . . 4 ⊢ Ⅎ𝑦𝐴
3		cbviunf.1	. . . . 5 ⊢ Ⅎ𝑦𝐵
4	3	nfcri 2891	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
5		cbviunf.2	. . . . 5 ⊢ Ⅎ𝑥𝐶
6	5	nfcri 2891	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶
7		cbviunf.3	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
8	7	eleq2d 2821	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶))
9	1, 2, 4, 6, 8	cbvrexfw 3289	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶)
10	9	abbii 2803	. 2 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶}
11		df-iun 4974	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
12		df-iun 4974	. 2 ⊢ ∪ 𝑦 ∈ 𝐴 𝐶 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶}
13	10, 11, 12	3eqtr4i 2769	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {cab 2714  Ⅎwnfc 2884  ∃wrex 3061  ∪ ciun 4972

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 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-12 2178  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-iun 4974

This theorem is referenced by:  aciunf1lem  32645