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Theorem nfdisj 4194
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 4184 . 2  |-  (Disj  x  e.  A B  <->  A. z E* x ( x  e.  A  /\  z  e.  B ) )
2 nftru 1563 . . . . 5  |-  F/ x  T.
3 nfcvf 2594 . . . . . . . 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 2587 . . . . . . 7  |-  ( -. 
A. y  y  =  x  ->  F/ y  x  e.  A )
7 nfdisj.2 . . . . . . . . 9  |-  F/_ y B
87nfcri 2566 . . . . . . . 8  |-  F/ y  z  e.  B
98a1i 11 . . . . . . 7  |-  ( -. 
A. y  y  =  x  ->  F/ y 
z  e.  B )
106, 9nfand 1843 . . . . . 6  |-  ( -. 
A. y  y  =  x  ->  F/ y
( x  e.  A  /\  z  e.  B
) )
1110adantl 453 . . . . 5  |-  ( (  T.  /\  -.  A. y  y  =  x
)  ->  F/ y
( x  e.  A  /\  z  e.  B
) )
122, 11nfmod2 2294 . . . 4  |-  (  T. 
->  F/ y E* x
( x  e.  A  /\  z  e.  B
) )
1312trud 1332 . . 3  |-  F/ y E* x ( x  e.  A  /\  z  e.  B )
1413nfal 1864 . 2  |-  F/ y A. z E* x
( x  e.  A  /\  z  e.  B
)
151, 14nfxfr 1579 1  |-  F/ yDisj  x  e.  A B
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    T. wtru 1325   A.wal 1549   F/wnf 1553    e. wcel 1725   E*wmo 2282   F/_wnfc 2559  Disj wdisj 4182
This theorem is referenced by:  disjxiun  4209
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rmo 2713  df-disj 4183
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