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Theorem sbcies 25998
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 5796 . . 3  |-  ( E `
 w )  e. 
_V
21a1i 11 . 2  |-  ( w  =  W  ->  ( E `  w )  e.  _V )
3 simpr 461 . . . . 5  |-  ( ( w  =  W  /\  a  =  ( E `  w ) )  -> 
a  =  ( E `
 w ) )
4 fveq2 5786 . . . . . . 7  |-  ( w  =  W  ->  ( E `  w )  =  ( E `  W ) )
5 sbcies.a . . . . . . 7  |-  A  =  ( E `  W
)
64, 5syl6reqr 2510 . . . . . 6  |-  ( w  =  W  ->  A  =  ( E `  w ) )
76adantr 465 . . . . 5  |-  ( ( w  =  W  /\  a  =  ( E `  w ) )  ->  A  =  ( E `  w ) )
83, 7eqtr4d 2494 . . . 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 3318 1  |-  ( w  =  W  ->  ( [. ( E `  w
)  /  a ]. ps 
<-> 
ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3065   [.wsbc 3281   ` cfv 5513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-nul 4516
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-iota 5476  df-fv 5521
This theorem is referenced by: (None)
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