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Theorem dfon2lem1 25412
Description: Lemma for dfon2 25421. (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 4318 . 2  |-  ( A. y  e.  { x  |  ( ph  /\  Tr  x  /\  ps ) } Tr  y  ->  Tr 
U. { x  |  ( ph  /\  Tr  x  /\  ps ) } )
2 nfsbc1v 3182 . . . . 5  |-  F/ x [. y  /  x ]. ph
3 nfv 1630 . . . . 5  |-  F/ x Tr  y
4 nfsbc1v 3182 . . . . 5  |-  F/ x [. y  /  x ]. ps
52, 3, 4nf3an 1850 . . . 4  |-  F/ x
( [. y  /  x ]. ph  /\  Tr  y  /\  [. y  /  x ]. ps )
6 vex 2961 . . . 4  |-  y  e. 
_V
7 sbceq1a 3173 . . . . 5  |-  ( x  =  y  ->  ( ph 
<-> 
[. y  /  x ]. ph ) )
8 treq 4310 . . . . 5  |-  ( x  =  y  ->  ( Tr  x  <->  Tr  y )
)
9 sbceq1a 3173 . . . . 5  |-  ( x  =  y  ->  ( ps 
<-> 
[. y  /  x ]. ps ) )
107, 8, 93anbi123d 1255 . . . 4  |-  ( x  =  y  ->  (
( ph  /\  Tr  x  /\  ps )  <->  ( [. y  /  x ]. ph  /\  Tr  y  /\  [. y  /  x ]. ps )
) )
115, 6, 10elabf 3083 . . 3  |-  ( y  e.  { x  |  ( ph  /\  Tr  x  /\  ps ) }  <-> 
( [. y  /  x ]. ph  /\  Tr  y  /\  [. y  /  x ]. ps ) )
1211simp2bi 974 . 2  |-  ( y  e.  { x  |  ( ph  /\  Tr  x  /\  ps ) }  ->  Tr  y )
131, 12mprg 2777 1  |-  Tr  U. { x  |  ( ph  /\  Tr  x  /\  ps ) }
Colors of variables: wff set class
Syntax hints:    /\ w3a 937    e. wcel 1726   {cab 2424   [.wsbc 3163   U.cuni 4017   Tr wtr 4304
This theorem is referenced by:  dfon2lem3  25414  dfon2lem7  25418
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-v 2960  df-sbc 3164  df-in 3329  df-ss 3336  df-uni 4018  df-iun 4097  df-tr 4305
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