Theorem cbviung 5038

Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. Usage of this theorem is discouraged because it depends on ax-13 2377. See cbviun 5036 for a version with more disjoint variable conditions, but not requiring ax-13 2377. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
cbviung	⊢ ∪ 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 cbviung

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbviung.1	. . . . 5 ⊢ Ⅎ𝑦𝐵
2	1	nfcri 2897	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
3		cbviung.2	. . . . 5 ⊢ Ⅎ𝑥𝐶
4	3	nfcri 2897	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶
5		cbviung.3	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
6	5	eleq2d 2827	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶))
7	2, 4, 6	cbvrex 3363	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶)
8	7	abbii 2809	. 2 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶}
9		df-iun 4993	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
10		df-iun 4993	. 2 ⊢ ∪ 𝑦 ∈ 𝐴 𝐶 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶}
11	8, 9, 10	3eqtr4i 2775	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  {cab 2714  Ⅎwnfc 2890  ∃wrex 3070  ∪ ciun 4991

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-10 2141  ax-11 2157  ax-12 2177  ax-13 2377  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-iun 4993

This theorem is referenced by:  cbviunvg  5042