Theorem nfii1 3679

Description: Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)

Assertion

Ref	Expression
nfii1	⊢ Ⅎx∩ x ∈ A B

Proof of Theorem nfii1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iin 3651	. 2 ⊢ ∩ x ∈ A B = {y ∣ ∀x ∈ A y ∈ B}
2		nfra1 2349	. . 3 ⊢ Ⅎx∀x ∈ A y ∈ B
3	2	nfab 2179	. 2 ⊢ Ⅎx{y ∣ ∀x ∈ A y ∈ B}
4	1, 3	nfcxfr 2172	1 ⊢ Ⅎx∩ 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-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:  dmiin  4523