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Theorem dfon2lem1 30380
Description: Lemma for dfon2 30389. (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 4475 . 2  |-  ( A. y  e.  { x  |  ( ph  /\  Tr  x  /\  ps ) } Tr  y  ->  Tr 
U. { x  |  ( ph  /\  Tr  x  /\  ps ) } )
2 nfsbc1v 3262 . . . . 5  |-  F/ x [. y  /  x ]. ph
3 nfv 1755 . . . . 5  |-  F/ x Tr  y
4 nfsbc1v 3262 . . . . 5  |-  F/ x [. y  /  x ]. ps
52, 3, 4nf3an 1990 . . . 4  |-  F/ x
( [. y  /  x ]. ph  /\  Tr  y  /\  [. y  /  x ]. ps )
6 vex 3025 . . . 4  |-  y  e. 
_V
7 sbceq1a 3253 . . . . 5  |-  ( x  =  y  ->  ( ph 
<-> 
[. y  /  x ]. ph ) )
8 treq 4467 . . . . 5  |-  ( x  =  y  ->  ( Tr  x  <->  Tr  y )
)
9 sbceq1a 3253 . . . . 5  |-  ( x  =  y  ->  ( ps 
<-> 
[. y  /  x ]. ps ) )
107, 8, 93anbi123d 1335 . . . 4  |-  ( x  =  y  ->  (
( ph  /\  Tr  x  /\  ps )  <->  ( [. y  /  x ]. ph  /\  Tr  y  /\  [. y  /  x ]. ps )
) )
115, 6, 10elabf 3159 . . 3  |-  ( y  e.  { x  |  ( ph  /\  Tr  x  /\  ps ) }  <-> 
( [. y  /  x ]. ph  /\  Tr  y  /\  [. y  /  x ]. ps ) )
1211simp2bi 1021 . 2  |-  ( y  e.  { x  |  ( ph  /\  Tr  x  /\  ps ) }  ->  Tr  y )
131, 12mprg 2728 1  |-  Tr  U. { x  |  ( ph  /\  Tr  x  /\  ps ) }
Colors of variables: wff setvar class
Syntax hints:    /\ w3a 982    e. wcel 1872   {cab 2414   [.wsbc 3242   U.cuni 4162   Tr wtr 4461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ral 2719  df-rex 2720  df-v 3024  df-sbc 3243  df-in 3386  df-ss 3393  df-uni 4163  df-iun 4244  df-tr 4462
This theorem is referenced by:  dfon2lem3  30382  dfon2lem7  30386
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