Theorem nfdisjv 4022

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 4012	. 2 |- (Disj x e. A B <-> A. z E* x ( x e. A /\ z e. B ) )
2		nfcv 2339	. . . . . 6 |- F/_ y x
3		nfdisjv.1	. . . . . 6 |- F/_ y A
4	2, 3	nfel 2348	. . . . 5 |- F/ y x e. A
5		nfdisjv.2	. . . . . 6 |- F/_ y B
6	5	nfcri 2333	. . . . 5 |- F/ y z e. B
7	4, 6	nfan 1579	. . . 4 |- F/ y ( x e. A /\ z e. B )
8	7	nfmo 2065	. . 3 |- F/ y E* x ( x e. A /\ z e. B )
9	8	nfal 1590	. 2 |- F/ y A. z E* x ( x e. A /\ z e. B )
10	1, 9	nfxfr 1488	1 |- F/ yDisj x e. A B

Syntax hints:   /\ wa 104   A.wal 1362   F/wnf 1474   E*wmo 2046   e. wcel 2167   F/_wnfc 2326  Disj wdisj 4010

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rmo 2483  df-disj 4011

This theorem is referenced by: (None)