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Theorem dfon2lem1 28820
Description: Lemma for dfon2 28829. (Contributed by Scott Fenton, 28-Feb-2011.)
Assertion
Ref Expression
dfon2lem1  |-  Tr  U. { x  |  ( ph  /\  Tr  x  /\  ps ) }

Proof of Theorem dfon2lem1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 truni 4554 . 2  |-  ( A. y  e.  { x  |  ( ph  /\  Tr  x  /\  ps ) } Tr  y  ->  Tr 
U. { x  |  ( ph  /\  Tr  x  /\  ps ) } )
2 nfsbc1v 3351 . . . . 5  |-  F/ x [. y  /  x ]. ph
3 nfv 1683 . . . . 5  |-  F/ x Tr  y
4 nfsbc1v 3351 . . . . 5  |-  F/ x [. y  /  x ]. ps
52, 3, 4nf3an 1877 . . . 4  |-  F/ x
( [. y  /  x ]. ph  /\  Tr  y  /\  [. y  /  x ]. ps )
6 vex 3116 . . . 4  |-  y  e. 
_V
7 sbceq1a 3342 . . . . 5  |-  ( x  =  y  ->  ( ph 
<-> 
[. y  /  x ]. ph ) )
8 treq 4546 . . . . 5  |-  ( x  =  y  ->  ( Tr  x  <->  Tr  y )
)
9 sbceq1a 3342 . . . . 5  |-  ( x  =  y  ->  ( ps 
<-> 
[. y  /  x ]. ps ) )
107, 8, 93anbi123d 1299 . . . 4  |-  ( x  =  y  ->  (
( ph  /\  Tr  x  /\  ps )  <->  ( [. y  /  x ]. ph  /\  Tr  y  /\  [. y  /  x ]. ps )
) )
115, 6, 10elabf 3249 . . 3  |-  ( y  e.  { x  |  ( ph  /\  Tr  x  /\  ps ) }  <-> 
( [. y  /  x ]. ph  /\  Tr  y  /\  [. y  /  x ]. ps ) )
1211simp2bi 1012 . 2  |-  ( y  e.  { x  |  ( ph  /\  Tr  x  /\  ps ) }  ->  Tr  y )
131, 12mprg 2827 1  |-  Tr  U. { x  |  ( ph  /\  Tr  x  /\  ps ) }
Colors of variables: wff setvar class
Syntax hints:    /\ w3a 973    e. wcel 1767   {cab 2452   [.wsbc 3331   U.cuni 4245   Tr wtr 4540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-v 3115  df-sbc 3332  df-in 3483  df-ss 3490  df-uni 4246  df-iun 4327  df-tr 4541
This theorem is referenced by:  dfon2lem3  28822  dfon2lem7  28826
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