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Theorem dfon2lem1 24210
Description: Lemma for dfon2 24219. (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 4143 . 2  |-  ( A. y  e.  { x  |  ( ph  /\  Tr  x  /\  ps ) } Tr  y  ->  Tr 
U. { x  |  ( ph  /\  Tr  x  /\  ps ) } )
2 nfsbc1v 3023 . . . . 5  |-  F/ x [. y  /  x ]. ph
3 nfv 1609 . . . . 5  |-  F/ x Tr  y
4 nfsbc1v 3023 . . . . 5  |-  F/ x [. y  /  x ]. ps
52, 3, 4nf3an 1786 . . . 4  |-  F/ x
( [. y  /  x ]. ph  /\  Tr  y  /\  [. y  /  x ]. ps )
6 vex 2804 . . . 4  |-  y  e. 
_V
7 sbceq1a 3014 . . . . 5  |-  ( x  =  y  ->  ( ph 
<-> 
[. y  /  x ]. ph ) )
8 treq 4135 . . . . 5  |-  ( x  =  y  ->  ( Tr  x  <->  Tr  y )
)
9 sbceq1a 3014 . . . . 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 2926 . . 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 2625 1  |-  Tr  U. { x  |  ( ph  /\  Tr  x  /\  ps ) }
Colors of variables: wff set class
Syntax hints:    /\ w3a 934    = wceq 1632    e. wcel 1696   {cab 2282   [.wsbc 3004   U.cuni 3843   Tr wtr 4129
This theorem is referenced by:  dfon2lem3  24212  dfon2lem7  24216
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-sbc 3005  df-in 3172  df-ss 3179  df-uni 3844  df-iun 3923  df-tr 4130
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