Theorem nfint 3880

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	|- F/_ x A

Assertion

Ref	Expression
nfint	|- F/_ x |^| A

Proof of Theorem nfint

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

Step	Hyp	Ref	Expression
1		dfint2 3872	. 2 |- |^| A = { y | A. z e. A y e. z }
2		nfint.1	. . . 4 |- F/_ x A
3		nfv 1539	. . . 4 |- F/ x y e. z
4	2, 3	nfralxy 2532	. . 3 |- F/ x A. z e. A y e. z
5	4	nfab 2341	. 2 |- F/_ x { y | A. z e. A y e. z }
6	1, 5	nfcxfr 2333	1 |- F/_ x |^| A

Syntax hints:   {cab 2179   F/_wnfc 2323   A.wral 2472   |^|cint 3870

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-int 3871

This theorem is referenced by: (None)