Theorem cbviin 3950

Description: Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.)

Hypotheses

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

Assertion

Ref	Expression
cbviin	⊢ ∩ 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 cbviin

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbviun.1	. . . . 5 ⊢ Ⅎ𝑦𝐵
2	1	nfcri 2330	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
3		cbviun.2	. . . . 5 ⊢ Ⅎ𝑥𝐶
4	3	nfcri 2330	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶
5		cbviun.3	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
6	5	eleq2d 2263	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶))
7	2, 4, 6	cbvral 2722	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑧 ∈ 𝐶)
8	7	abbii 2309	. 2 ⊢ {𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = {𝑧 ∣ ∀𝑦 ∈ 𝐴 𝑧 ∈ 𝐶}
9		df-iin 3915	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
10		df-iin 3915	. 2 ⊢ ∩ 𝑦 ∈ 𝐴 𝐶 = {𝑧 ∣ ∀𝑦 ∈ 𝐴 𝑧 ∈ 𝐶}
11	8, 9, 10	3eqtr4i 2224	1 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2164  {cab 2179  Ⅎwnfc 2323  ∀wral 2472  ∩ ciin 3913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-iin 3915

This theorem is referenced by:  cbviinv  3952