Theorem nfdisjv 4018

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 4008	. 2 |- (Disj x e. A B <-> A. z E* x ( x e. A /\ z e. B ) )
2		nfcv 2336	. . . . . 6 |- F/_ y x
3		nfdisjv.1	. . . . . 6 |- F/_ y A
4	2, 3	nfel 2345	. . . . 5 |- F/ y x e. A
5		nfdisjv.2	. . . . . 6 |- F/_ y B
6	5	nfcri 2330	. . . . 5 |- F/ y z e. B
7	4, 6	nfan 1576	. . . 4 |- F/ y ( x e. A /\ z e. B )
8	7	nfmo 2062	. . 3 |- F/ y E* x ( x e. A /\ z e. B )
9	8	nfal 1587	. 2 |- F/ y A. z E* x ( x e. A /\ z e. B )
10	1, 9	nfxfr 1485	1 |- F/ yDisj x e. A B

Syntax hints:   /\ wa 104   A.wal 1362   F/wnf 1471   E*wmo 2043   e. wcel 2164   F/_wnfc 2323  Disj wdisj 4006

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rmo 2480  df-disj 4007

This theorem is referenced by: (None)