Theorem nfint 3646

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 3638	. 2 |- |^| A = { y | A. z e. A y e. z }
2		nfint.1	. . . 4 |- F/_ x A
3		nfv 1461	. . . 4 |- F/ x y e. z
4	2, 3	nfralxy 2402	. . 3 |- F/ x A. z e. A y e. z
5	4	nfab 2223	. 2 |- F/_ x { y | A. z e. A y e. z }
6	1, 5	nfcxfr 2216	1 |- F/_ x |^| A

Syntax hints:   {cab 2067   F/_wnfc 2206   A.wral 2348   |^|cint 3636

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-int 3637

This theorem is referenced by: (None)