Theorem nfdisjv 3856

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 3846	. 2 |- (Disj x e. A B <-> A. z E* x ( x e. A /\ z e. B ) )
2		nfcv 2235	. . . . . 6 |- F/_ y x
3		nfdisjv.1	. . . . . 6 |- F/_ y A
4	2, 3	nfel 2244	. . . . 5 |- F/ y x e. A
5		nfdisjv.2	. . . . . 6 |- F/_ y B
6	5	nfcri 2229	. . . . 5 |- F/ y z e. B
7	4, 6	nfan 1509	. . . 4 |- F/ y ( x e. A /\ z e. B )
8	7	nfmo 1975	. . 3 |- F/ y E* x ( x e. A /\ z e. B )
9	8	nfal 1520	. 2 |- F/ y A. z E* x ( x e. A /\ z e. B )
10	1, 9	nfxfr 1415	1 |- F/ yDisj x e. A B

Syntax hints:   /\ wa 103   A.wal 1294   F/wnf 1401   e. wcel 1445   E*wmo 1956   F/_wnfc 2222  Disj wdisj 3844

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rmo 2378  df-disj 3845

This theorem is referenced by: (None)