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Theorem nfdisj 4385
Description: Bound-variable hypothesis builder for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
nfdisj.1  |-  F/_ y A
nfdisj.2  |-  F/_ y B
Assertion
Ref Expression
nfdisj  |-  F/ yDisj  x  e.  A  B

Proof of Theorem nfdisj
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dfdisj2 4375 . 2  |-  (Disj  x  e.  A  B  <->  A. z E* x ( x  e.  A  /\  z  e.  B ) )
2 nftru 1677 . . . . 5  |-  F/ x T.
3 nfcvf 2615 . . . . . . . 8  |-  ( -. 
A. y  y  =  x  ->  F/_ y x )
4 nfdisj.1 . . . . . . . . 9  |-  F/_ y A
54a1i 11 . . . . . . . 8  |-  ( -. 
A. y  y  =  x  ->  F/_ y A )
63, 5nfeld 2600 . . . . . . 7  |-  ( -. 
A. y  y  =  x  ->  F/ y  x  e.  A )
7 nfdisj.2 . . . . . . . . 9  |-  F/_ y B
87nfcri 2586 . . . . . . . 8  |-  F/ y  z  e.  B
98a1i 11 . . . . . . 7  |-  ( -. 
A. y  y  =  x  ->  F/ y 
z  e.  B )
106, 9nfand 2008 . . . . . 6  |-  ( -. 
A. y  y  =  x  ->  F/ y
( x  e.  A  /\  z  e.  B
) )
1110adantl 468 . . . . 5  |-  ( ( T.  /\  -.  A. y  y  =  x
)  ->  F/ y
( x  e.  A  /\  z  e.  B
) )
122, 11nfmod2 2312 . . . 4  |-  ( T. 
->  F/ y E* x
( x  e.  A  /\  z  e.  B
) )
1312trud 1453 . . 3  |-  F/ y E* x ( x  e.  A  /\  z  e.  B )
1413nfal 2030 . 2  |-  F/ y A. z E* x
( x  e.  A  /\  z  e.  B
)
151, 14nfxfr 1696 1  |-  F/ yDisj  x  e.  A  B
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 371   A.wal 1442   T. wtru 1445   F/wnf 1667    e. wcel 1887   E*wmo 2300   F/_wnfc 2579  Disj wdisj 4373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-cleq 2444  df-clel 2447  df-nfc 2581  df-rmo 2745  df-disj 4374
This theorem is referenced by:  disjxiun  4399
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