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Theorem rusgranumwlklem0 25153
Description: Lemma 0 for rusgranumwlk 25162. (Contributed by Alexander van der Vekens, 23-Aug-2018.)
Assertion
Ref Expression
rusgranumwlklem0  |-  ( Y  e.  { w  e.  Z  |  ( w `
 0 )  =  P }  ->  { w  e.  X  |  ( ph  /\  ps ) }  =  { w  e.  X  |  ( ph  /\  ( Y `  0
)  =  P  /\  ps ) } )
Distinct variable groups:    w, P    w, Y    w, Z
Allowed substitution hints:    ph( w)    ps( w)    X( w)

Proof of Theorem rusgranumwlklem0
StepHypRef Expression
1 fveq1 5847 . . . 4  |-  ( w  =  Y  ->  (
w `  0 )  =  ( Y ` 
0 ) )
21eqeq1d 2456 . . 3  |-  ( w  =  Y  ->  (
( w `  0
)  =  P  <->  ( Y `  0 )  =  P ) )
32elrab 3254 . 2  |-  ( Y  e.  { w  e.  Z  |  ( w `
 0 )  =  P }  <->  ( Y  e.  Z  /\  ( Y `  0 )  =  P ) )
4 ibar 502 . . . . 5  |-  ( ( Y `  0 )  =  P  ->  (
( ph  /\  ps )  <->  ( ( Y `  0
)  =  P  /\  ( ph  /\  ps )
) ) )
5 3anass 975 . . . . . 6  |-  ( ( ( Y `  0
)  =  P  /\  ph 
/\  ps )  <->  ( ( Y `  0 )  =  P  /\  ( ph  /\  ps ) ) )
6 3ancoma 978 . . . . . 6  |-  ( ( ( Y `  0
)  =  P  /\  ph 
/\  ps )  <->  ( ph  /\  ( Y `  0
)  =  P  /\  ps ) )
75, 6bitr3i 251 . . . . 5  |-  ( ( ( Y `  0
)  =  P  /\  ( ph  /\  ps )
)  <->  ( ph  /\  ( Y `  0 )  =  P  /\  ps ) )
84, 7syl6bb 261 . . . 4  |-  ( ( Y `  0 )  =  P  ->  (
( ph  /\  ps )  <->  (
ph  /\  ( Y `  0 )  =  P  /\  ps )
) )
98ad2antlr 724 . . 3  |-  ( ( ( Y  e.  Z  /\  ( Y `  0
)  =  P )  /\  w  e.  X
)  ->  ( ( ph  /\  ps )  <->  ( ph  /\  ( Y `  0
)  =  P  /\  ps ) ) )
109rabbidva 3097 . 2  |-  ( ( Y  e.  Z  /\  ( Y `  0 )  =  P )  ->  { w  e.  X  |  ( ph  /\  ps ) }  =  {
w  e.  X  | 
( ph  /\  ( Y `  0 )  =  P  /\  ps ) } )
113, 10sylbi 195 1  |-  ( Y  e.  { w  e.  Z  |  ( w `
 0 )  =  P }  ->  { w  e.  X  |  ( ph  /\  ps ) }  =  { w  e.  X  |  ( ph  /\  ( Y `  0
)  =  P  /\  ps ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   {crab 2808   ` cfv 5570   0cc0 9481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578
This theorem is referenced by:  rusgranumwlks  25161
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