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Theorem sbcies 27579
Description: A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019.)
Hypotheses
Ref Expression
sbcies.a  |-  A  =  ( E `  W
)
sbcies.1  |-  ( a  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
sbcies  |-  ( w  =  W  ->  ( [. ( E `  w
)  /  a ]. ps 
<-> 
ph ) )
Distinct variable groups:    w, a    E, a    W, a    ph, a
Allowed substitution hints:    ph( w)    ps( w, a)    A( w, a)    E( w)    W( w)

Proof of Theorem sbcies
StepHypRef Expression
1 fvex 5858 . . 3  |-  ( E `
 w )  e. 
_V
21a1i 11 . 2  |-  ( w  =  W  ->  ( E `  w )  e.  _V )
3 simpr 459 . . . . 5  |-  ( ( w  =  W  /\  a  =  ( E `  w ) )  -> 
a  =  ( E `
 w ) )
4 fveq2 5848 . . . . . . 7  |-  ( w  =  W  ->  ( E `  w )  =  ( E `  W ) )
5 sbcies.a . . . . . . 7  |-  A  =  ( E `  W
)
64, 5syl6reqr 2514 . . . . . 6  |-  ( w  =  W  ->  A  =  ( E `  w ) )
76adantr 463 . . . . 5  |-  ( ( w  =  W  /\  a  =  ( E `  w ) )  ->  A  =  ( E `  w ) )
83, 7eqtr4d 2498 . . . 4  |-  ( ( w  =  W  /\  a  =  ( E `  w ) )  -> 
a  =  A )
9 sbcies.1 . . . 4  |-  ( a  =  A  ->  ( ph 
<->  ps ) )
108, 9syl 16 . . 3  |-  ( ( w  =  W  /\  a  =  ( E `  w ) )  -> 
( ph  <->  ps ) )
1110bicomd 201 . 2  |-  ( ( w  =  W  /\  a  =  ( E `  w ) )  -> 
( ps  <->  ph ) )
122, 11sbcied 3361 1  |-  ( w  =  W  ->  ( [. ( E `  w
)  /  a ]. ps 
<-> 
ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   [.wsbc 3324   ` cfv 5570
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  ax-nul 4568
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578
This theorem is referenced by: (None)
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