Theorem nfiinxy 3675

Description: Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)

Hypotheses

Ref	Expression
nfiunxy.1	⊢ ℲyA
nfiunxy.2	⊢ ℲyB

Assertion

Ref	Expression
nfiinxy	⊢ Ⅎy∩ x ∈ A B

Distinct variable group:   x,y

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

Proof of Theorem nfiinxy

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iin 3651	. 2 ⊢ ∩ x ∈ A B = {z ∣ ∀x ∈ A z ∈ B}
2		nfiunxy.1	. . . 4 ⊢ ℲyA
3		nfiunxy.2	. . . . 5 ⊢ ℲyB
4	3	nfcri 2169	. . . 4 ⊢ Ⅎy z ∈ B
5	2, 4	nfralxy 2354	. . 3 ⊢ Ⅎy∀x ∈ A z ∈ B
6	5	nfab 2179	. 2 ⊢ Ⅎy{z ∣ ∀x ∈ A z ∈ B}
7	1, 6	nfcxfr 2172	1 ⊢ Ⅎy∩ x ∈ A B

Syntax hints:   ∈ wcel 1390  {cab 2023  Ⅎwnfc 2162  ∀wral 2300  ∩ ciin 3649

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-iin 3651

This theorem is referenced by:  iinab  3709