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Theorem dfon2lem1 24139
Description: Lemma for dfon2 24148. (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 4127 . 2  |-  ( A. y  e.  { x  |  ( ph  /\  Tr  x  /\  ps ) } Tr  y  ->  Tr 
U. { x  |  ( ph  /\  Tr  x  /\  ps ) } )
2 nfsbc1v 3010 . . . . 5  |-  F/ x [. y  /  x ]. ph
3 nfv 1605 . . . . 5  |-  F/ x Tr  y
4 nfsbc1v 3010 . . . . 5  |-  F/ x [. y  /  x ]. ps
52, 3, 4nf3an 1774 . . . 4  |-  F/ x
( [. y  /  x ]. ph  /\  Tr  y  /\  [. y  /  x ]. ps )
6 vex 2791 . . . 4  |-  y  e. 
_V
7 sbceq1a 3001 . . . . 5  |-  ( x  =  y  ->  ( ph 
<-> 
[. y  /  x ]. ph ) )
8 treq 4119 . . . . 5  |-  ( x  =  y  ->  ( Tr  x  <->  Tr  y )
)
9 sbceq1a 3001 . . . . 5  |-  ( x  =  y  ->  ( ps 
<-> 
[. y  /  x ]. ps ) )
107, 8, 93anbi123d 1252 . . . 4  |-  ( x  =  y  ->  (
( ph  /\  Tr  x  /\  ps )  <->  ( [. y  /  x ]. ph  /\  Tr  y  /\  [. y  /  x ]. ps )
) )
115, 6, 10elabf 2913 . . 3  |-  ( y  e.  { x  |  ( ph  /\  Tr  x  /\  ps ) }  <-> 
( [. y  /  x ]. ph  /\  Tr  y  /\  [. y  /  x ]. ps ) )
1211simp2bi 971 . 2  |-  ( y  e.  { x  |  ( ph  /\  Tr  x  /\  ps ) }  ->  Tr  y )
131, 12mprg 2612 1  |-  Tr  U. { x  |  ( ph  /\  Tr  x  /\  ps ) }
Colors of variables: wff set class
Syntax hints:    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269   [.wsbc 2991   U.cuni 3827   Tr wtr 4113
This theorem is referenced by:  dfon2lem3  24141  dfon2lem7  24145
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-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-sbc 2992  df-in 3159  df-ss 3166  df-uni 3828  df-iun 3907  df-tr 4114
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