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Theorem sbcie3s 14689
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 5882 . . 3  |-  ( E `
 w )  e. 
_V
21a1i 11 . 2  |-  ( w  =  W  ->  ( E `  w )  e.  _V )
3 fvex 5882 . . . 4  |-  ( F `
 w )  e. 
_V
43a1i 11 . . 3  |-  ( ( w  =  W  /\  a  =  ( E `  w ) )  -> 
( F `  w
)  e.  _V )
5 fvex 5882 . . . . 5  |-  ( G `
 w )  e. 
_V
65a1i 11 . . . 4  |-  ( ( ( w  =  W  /\  a  =  ( E `  w ) )  /\  b  =  ( F `  w
) )  ->  ( G `  w )  e.  _V )
7 simpllr 760 . . . . . . . 8  |-  ( ( ( ( w  =  W  /\  a  =  ( E `  w
) )  /\  b  =  ( F `  w ) )  /\  c  =  ( G `  w ) )  -> 
a  =  ( E `
 w ) )
8 fveq2 5872 . . . . . . . . 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 2498 . . . . . . 7  |-  ( ( ( ( w  =  W  /\  a  =  ( E `  w
) )  /\  b  =  ( F `  w ) )  /\  c  =  ( G `  w ) )  -> 
a  =  ( E `
 W ) )
11 sbcie3s.a . . . . . . 7  |-  A  =  ( E `  W
)
1210, 11syl6eqr 2516 . . . . . 6  |-  ( ( ( ( w  =  W  /\  a  =  ( E `  w
) )  /\  b  =  ( F `  w ) )  /\  c  =  ( G `  w ) )  -> 
a  =  A )
13 simplr 755 . . . . . . . 8  |-  ( ( ( ( w  =  W  /\  a  =  ( E `  w
) )  /\  b  =  ( F `  w ) )  /\  c  =  ( G `  w ) )  -> 
b  =  ( F `
 w ) )
14 fveq2 5872 . . . . . . . . 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 2498 . . . . . . 7  |-  ( ( ( ( w  =  W  /\  a  =  ( E `  w
) )  /\  b  =  ( F `  w ) )  /\  c  =  ( G `  w ) )  -> 
b  =  ( F `
 W ) )
17 sbcie3s.b . . . . . . 7  |-  B  =  ( F `  W
)
1816, 17syl6eqr 2516 . . . . . 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 5872 . . . . . . . . 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 2498 . . . . . . 7  |-  ( ( ( ( w  =  W  /\  a  =  ( E `  w
) )  /\  b  =  ( F `  w ) )  /\  c  =  ( G `  w ) )  -> 
c  =  ( G `
 W ) )
23 sbcie3s.c . . . . . . 7  |-  C  =  ( G `  W
)
2422, 23syl6eqr 2516 . . . . . 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 1228 . . . . 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 3364 . . 3  |-  ( ( ( w  =  W  /\  a  =  ( E `  w ) )  /\  b  =  ( F `  w
) )  ->  ( [. ( G `  w
)  /  c ]. ps 
<-> 
ph ) )
294, 28sbcied 3364 . 2  |-  ( ( w  =  W  /\  a  =  ( E `  w ) )  -> 
( [. ( F `  w )  /  b ]. [. ( G `  w )  /  c ]. ps  <->  ph ) )
302, 29sbcied 3364 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 973    = wceq 1395    e. wcel 1819   _Vcvv 3109   [.wsbc 3327   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-nul 4586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602
This theorem is referenced by:  istrkgcb  23978  istrkg2d  23981  istrkgld  23982  legval  24096  afsval  28726
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