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Theorem sbcie3s 14223
Description: A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 15-Mar-2019.)
Hypotheses
Ref Expression
sbcie3s.a  |-  A  =  ( E `  W
)
sbcie3s.b  |-  B  =  ( F `  W
)
sbcie3s.c  |-  C  =  ( G `  W
)
sbcie3s.1  |-  ( ( a  =  A  /\  b  =  B  /\  c  =  C )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
sbcie3s  |-  ( w  =  W  ->  ( [. ( E `  w
)  /  a ]. [. ( F `  w
)  /  b ]. [. ( G `  w
)  /  c ]. ps 
<-> 
ph ) )
Distinct variable groups:    a, b,
c, w    E, a,
b, c    F, b,
c    G, c    W, a, b, c    ph, a,
b, c
Allowed substitution hints:    ph( w)    ps( w, a, b, c)    A( w, a, b, c)    B( w, a, b, c)    C( w, a, b, c)    E( w)    F( w, a)    G( w, a, b)    W( w)

Proof of Theorem sbcie3s
StepHypRef Expression
1 fvex 5706 . . 3  |-  ( E `
 w )  e. 
_V
21a1i 11 . 2  |-  ( w  =  W  ->  ( E `  w )  e.  _V )
3 fvex 5706 . . . 4  |-  ( F `
 w )  e. 
_V
43a1i 11 . . 3  |-  ( ( w  =  W  /\  a  =  ( E `  w ) )  -> 
( F `  w
)  e.  _V )
5 fvex 5706 . . . . 5  |-  ( G `
 w )  e. 
_V
65a1i 11 . . . 4  |-  ( ( ( w  =  W  /\  a  =  ( E `  w ) )  /\  b  =  ( F `  w
) )  ->  ( G `  w )  e.  _V )
7 simpllr 758 . . . . . . . 8  |-  ( ( ( ( w  =  W  /\  a  =  ( E `  w
) )  /\  b  =  ( F `  w ) )  /\  c  =  ( G `  w ) )  -> 
a  =  ( E `
 w ) )
8 fveq2 5696 . . . . . . . . 9  |-  ( w  =  W  ->  ( E `  w )  =  ( E `  W ) )
98ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( w  =  W  /\  a  =  ( E `  w
) )  /\  b  =  ( F `  w ) )  /\  c  =  ( G `  w ) )  -> 
( E `  w
)  =  ( E `
 W ) )
107, 9eqtrd 2475 . . . . . . 7  |-  ( ( ( ( w  =  W  /\  a  =  ( E `  w
) )  /\  b  =  ( F `  w ) )  /\  c  =  ( G `  w ) )  -> 
a  =  ( E `
 W ) )
11 sbcie3s.a . . . . . . 7  |-  A  =  ( E `  W
)
1210, 11syl6eqr 2493 . . . . . 6  |-  ( ( ( ( w  =  W  /\  a  =  ( E `  w
) )  /\  b  =  ( F `  w ) )  /\  c  =  ( G `  w ) )  -> 
a  =  A )
13 simplr 754 . . . . . . . 8  |-  ( ( ( ( w  =  W  /\  a  =  ( E `  w
) )  /\  b  =  ( F `  w ) )  /\  c  =  ( G `  w ) )  -> 
b  =  ( F `
 w ) )
14 fveq2 5696 . . . . . . . . 9  |-  ( w  =  W  ->  ( F `  w )  =  ( F `  W ) )
1514ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( w  =  W  /\  a  =  ( E `  w
) )  /\  b  =  ( F `  w ) )  /\  c  =  ( G `  w ) )  -> 
( F `  w
)  =  ( F `
 W ) )
1613, 15eqtrd 2475 . . . . . . 7  |-  ( ( ( ( w  =  W  /\  a  =  ( E `  w
) )  /\  b  =  ( F `  w ) )  /\  c  =  ( G `  w ) )  -> 
b  =  ( F `
 W ) )
17 sbcie3s.b . . . . . . 7  |-  B  =  ( F `  W
)
1816, 17syl6eqr 2493 . . . . . 6  |-  ( ( ( ( w  =  W  /\  a  =  ( E `  w
) )  /\  b  =  ( F `  w ) )  /\  c  =  ( G `  w ) )  -> 
b  =  B )
19 simpr 461 . . . . . . . 8  |-  ( ( ( ( w  =  W  /\  a  =  ( E `  w
) )  /\  b  =  ( F `  w ) )  /\  c  =  ( G `  w ) )  -> 
c  =  ( G `
 w ) )
20 fveq2 5696 . . . . . . . . 9  |-  ( w  =  W  ->  ( G `  w )  =  ( G `  W ) )
2120ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( w  =  W  /\  a  =  ( E `  w
) )  /\  b  =  ( F `  w ) )  /\  c  =  ( G `  w ) )  -> 
( G `  w
)  =  ( G `
 W ) )
2219, 21eqtrd 2475 . . . . . . 7  |-  ( ( ( ( w  =  W  /\  a  =  ( E `  w
) )  /\  b  =  ( F `  w ) )  /\  c  =  ( G `  w ) )  -> 
c  =  ( G `
 W ) )
23 sbcie3s.c . . . . . . 7  |-  C  =  ( G `  W
)
2422, 23syl6eqr 2493 . . . . . 6  |-  ( ( ( ( w  =  W  /\  a  =  ( E `  w
) )  /\  b  =  ( F `  w ) )  /\  c  =  ( G `  w ) )  -> 
c  =  C )
25 sbcie3s.1 . . . . . 6  |-  ( ( a  =  A  /\  b  =  B  /\  c  =  C )  ->  ( ph  <->  ps )
)
2612, 18, 24, 25syl3anc 1218 . . . . 5  |-  ( ( ( ( w  =  W  /\  a  =  ( E `  w
) )  /\  b  =  ( F `  w ) )  /\  c  =  ( G `  w ) )  -> 
( ph  <->  ps ) )
2726bicomd 201 . . . 4  |-  ( ( ( ( w  =  W  /\  a  =  ( E `  w
) )  /\  b  =  ( F `  w ) )  /\  c  =  ( G `  w ) )  -> 
( ps  <->  ph ) )
286, 27sbcied 3228 . . 3  |-  ( ( ( w  =  W  /\  a  =  ( E `  w ) )  /\  b  =  ( F `  w
) )  ->  ( [. ( G `  w
)  /  c ]. ps 
<-> 
ph ) )
294, 28sbcied 3228 . 2  |-  ( ( w  =  W  /\  a  =  ( E `  w ) )  -> 
( [. ( F `  w )  /  b ]. [. ( G `  w )  /  c ]. ps  <->  ph ) )
302, 29sbcied 3228 1  |-  ( w  =  W  ->  ( [. ( E `  w
)  /  a ]. [. ( F `  w
)  /  b ]. [. ( G `  w
)  /  c ]. ps 
<-> 
ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2977   [.wsbc 3191   ` cfv 5423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-nul 4426
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-iota 5386  df-fv 5431
This theorem is referenced by:  istrkgcb  22924  istrkg2d  22927  legval  23020  afsval  27000
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