Theorem nfdisjv 4081

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 4071	. 2 |- (Disj x e. A B <-> A. z E* x ( x e. A /\ z e. B ) )
2		nfcv 2375	. . . . . 6 |- F/_ y x
3		nfdisjv.1	. . . . . 6 |- F/_ y A
4	2, 3	nfel 2384	. . . . 5 |- F/ y x e. A
5		nfdisjv.2	. . . . . 6 |- F/_ y B
6	5	nfcri 2369	. . . . 5 |- F/ y z e. B
7	4, 6	nfan 1614	. . . 4 |- F/ y ( x e. A /\ z e. B )
8	7	nfmo 2099	. . 3 |- F/ y E* x ( x e. A /\ z e. B )
9	8	nfal 1625	. 2 |- F/ y A. z E* x ( x e. A /\ z e. B )
10	1, 9	nfxfr 1523	1 |- F/ yDisj x e. A B

Syntax hints:   /\ wa 104   A.wal 1396   F/wnf 1509   E*wmo 2080   e. wcel 2202   F/_wnfc 2362  Disj wdisj 4069

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rmo 2519  df-disj 4070

This theorem is referenced by: (None)