Theorem nfdisjv 4047

Description: Bound-variable hypothesis builder for disjoint collection. (Contributed by Jim Kingdon, 19-Aug-2018.)

Hypotheses

Ref	Expression
nfdisjv.1	|- F/_ y A
nfdisjv.2	|- F/_ y B

Assertion

Ref	Expression
nfdisjv	|- F/ yDisj x e. A B

Distinct variable group:   x, y

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

Proof of Theorem nfdisjv

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfdisj2 4037	. 2 |- (Disj x e. A B <-> A. z E* x ( x e. A /\ z e. B ) )
2		nfcv 2350	. . . . . 6 |- F/_ y x
3		nfdisjv.1	. . . . . 6 |- F/_ y A
4	2, 3	nfel 2359	. . . . 5 |- F/ y x e. A
5		nfdisjv.2	. . . . . 6 |- F/_ y B
6	5	nfcri 2344	. . . . 5 |- F/ y z e. B
7	4, 6	nfan 1589	. . . 4 |- F/ y ( x e. A /\ z e. B )
8	7	nfmo 2075	. . 3 |- F/ y E* x ( x e. A /\ z e. B )
9	8	nfal 1600	. 2 |- F/ y A. z E* x ( x e. A /\ z e. B )
10	1, 9	nfxfr 1498	1 |- F/ yDisj x e. A B

Syntax hints:   /\ wa 104   A.wal 1371   F/wnf 1484   E*wmo 2056   e. wcel 2178   F/_wnfc 2337  Disj wdisj 4035

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rmo 2494  df-disj 4036

This theorem is referenced by: (None)