Theorem cbviing 5003

Description: Change bound variables in an indexed intersection. Usage of this theorem is discouraged because it depends on ax-13 2404. See cbviin 5001 for a version with more disjoint variable conditions, but not requiring ax-13 2404. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
cbviing	⊢ ∩ 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 cbviing

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbviung.1	. . . . 5 ⊢ Ⅎ𝑦𝐵
2	1	nfcri 2917	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
3		cbviung.2	. . . . 5 ⊢ Ⅎ𝑥𝐶
4	3	nfcri 2917	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶
5		cbviung.3	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
6	5	eleq2d 2849	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶))
7	2, 4, 6	cbvral 3351	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝑧 ∈ 𝐶)
8	7	abbii 2830	. 2 ⊢ {𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = {𝑧 ∣ ∀𝑦 ∈ 𝐴 𝑧 ∈ 𝐶}
9		df-iin 4960	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
10		df-iin 4960	. 2 ⊢ ∩ 𝑦 ∈ 𝐴 𝐶 = {𝑧 ∣ ∀𝑦 ∈ 𝐴 𝑧 ∈ 𝐶}
11	8, 9, 10	3eqtr4i 2796	1 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  {cab 2741  Ⅎwnfc 2910  ∀wral 3079  ∩ ciin 4958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-13 2404  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-iin 4960

This theorem is referenced by:  cbviinvg  5007