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Theorem nfdisj 4005
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 df-disj 3994 . . 3  |-  (Disj  x  e.  A B  <->  A. z E* x  e.  A
z  e.  B )
2 df-rmo 2551 . . . 4  |-  ( E* x  e.  A z  e.  B  <->  E* x
( x  e.  A  /\  z  e.  B
) )
32albii 1553 . . 3  |-  ( A. z E* x  e.  A
z  e.  B  <->  A. z E* x ( x  e.  A  /\  z  e.  B ) )
41, 3bitri 240 . 2  |-  (Disj  x  e.  A B  <->  A. z E* x ( x  e.  A  /\  z  e.  B ) )
5 nftru 1541 . . . . 5  |-  F/ x  T.
6 nfcvf 2441 . . . . . . . 8  |-  ( -. 
A. y  y  =  x  ->  F/_ y x )
7 nfdisj.1 . . . . . . . . 9  |-  F/_ y A
87a1i 10 . . . . . . . 8  |-  ( -. 
A. y  y  =  x  ->  F/_ y A )
96, 8nfeld 2434 . . . . . . 7  |-  ( -. 
A. y  y  =  x  ->  F/ y  x  e.  A )
10 nfdisj.2 . . . . . . . . 9  |-  F/_ y B
1110nfcri 2413 . . . . . . . 8  |-  F/ y  z  e.  B
1211a1i 10 . . . . . . 7  |-  ( -. 
A. y  y  =  x  ->  F/ y 
z  e.  B )
139, 12nfand 1763 . . . . . 6  |-  ( -. 
A. y  y  =  x  ->  F/ y
( x  e.  A  /\  z  e.  B
) )
1413adantl 452 . . . . 5  |-  ( (  T.  /\  -.  A. y  y  =  x
)  ->  F/ y
( x  e.  A  /\  z  e.  B
) )
155, 14nfmod2 2156 . . . 4  |-  (  T. 
->  F/ y E* x
( x  e.  A  /\  z  e.  B
) )
1615trud 1314 . . 3  |-  F/ y E* x ( x  e.  A  /\  z  e.  B )
1716nfal 1766 . 2  |-  F/ y A. z E* x
( x  e.  A  /\  z  e.  B
)
184, 17nfxfr 1557 1  |-  F/ yDisj  x  e.  A B
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    T. wtru 1307   A.wal 1527   F/wnf 1531    e. wcel 1684   E*wmo 2144   F/_wnfc 2406   E*wrmo 2546  Disj wdisj 3993
This theorem is referenced by:  disjxiun  4020
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rmo 2551  df-disj 3994
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