Theorem nfint 3937

Description: Bound-variable hypothesis builder for intersection. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Hypothesis

Ref	Expression
nfint.1	⊢ ℲxA

Assertion

Ref	Expression
nfint	⊢ Ⅎx∩A

Proof of Theorem nfint

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfint2 3929	. 2 ⊢ ∩A = {y ∣ ∀z ∈ A y ∈ z}
2		nfint.1	. . . 4 ⊢ ℲxA
3		nfv 1619	. . . 4 ⊢ Ⅎx y ∈ z
4	2, 3	nfral 2668	. . 3 ⊢ Ⅎx∀z ∈ A y ∈ z
5	4	nfab 2494	. 2 ⊢ Ⅎx{y ∣ ∀z ∈ A y ∈ z}
6	1, 5	nfcxfr 2487	1 ⊢ Ⅎx∩A

Syntax hints:   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2477  ∀wral 2615  ∩cint 3927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-int 3928

This theorem is referenced by: (None)