Theorem cbviunf 33204

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 2917	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
5		cbviunf.2	. . . . 5 ⊢ Ⅎ𝑥𝐶
6	5	nfcri 2917	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶
7		cbviunf.3	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
8	7	eleq2d 2849	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶))
9	1, 2, 4, 6, 8	cbvrexfw 3306	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶)
10	9	abbii 2830	. 2 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶}
11		df-iun 4955	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
12		df-iun 4955	. 2 ⊢ ∪ 𝑦 ∈ 𝐴 𝐶 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶}
13	10, 11, 12	3eqtr4i 2796	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2146  {cab 2741  Ⅎwnfc 2910  ∃wrex 3089  ∪ ciun 4953

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-11 2196  ax-12 2215  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-iun 4955

This theorem is used by:  aciunf1lem  33310