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Theorem sbcie2s 14216
Description: A special version of class substitution commonly used for structures. (Contributed by Thierry Arnoux, 14-Mar-2019.)
Hypotheses
Ref Expression
sbcie2s.a  |-  A  =  ( E `  W
)
sbcie2s.b  |-  B  =  ( F `  W
)
sbcie2s.1  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
sbcie2s  |-  ( w  =  W  ->  ( [. ( E `  w
)  /  a ]. [. ( F `  w
)  /  b ]. ps 
<-> 
ph ) )
Distinct variable groups:    a, b, w    E, a, b    F, b    W, a, b    ph, a,
b
Allowed substitution hints:    ph( w)    ps( w, a, b)    A( w, a, b)    B( w, a, b)    E( w)    F( w, a)    W( w)

Proof of Theorem sbcie2s
StepHypRef Expression
1 fvex 5700 . 2  |-  ( E `
 w )  e. 
_V
2 fvex 5700 . 2  |-  ( F `
 w )  e. 
_V
3 simprl 755 . . . . . . 7  |-  ( ( w  =  W  /\  ( a  =  ( E `  w )  /\  b  =  ( F `  w ) ) )  ->  a  =  ( E `  w ) )
4 fveq2 5690 . . . . . . . . 9  |-  ( w  =  W  ->  ( E `  w )  =  ( E `  W ) )
5 sbcie2s.a . . . . . . . . 9  |-  A  =  ( E `  W
)
64, 5syl6eqr 2492 . . . . . . . 8  |-  ( w  =  W  ->  ( E `  w )  =  A )
76adantr 465 . . . . . . 7  |-  ( ( w  =  W  /\  ( a  =  ( E `  w )  /\  b  =  ( F `  w ) ) )  ->  ( E `  w )  =  A )
83, 7eqtrd 2474 . . . . . 6  |-  ( ( w  =  W  /\  ( a  =  ( E `  w )  /\  b  =  ( F `  w ) ) )  ->  a  =  A )
9 simprr 756 . . . . . . 7  |-  ( ( w  =  W  /\  ( a  =  ( E `  w )  /\  b  =  ( F `  w ) ) )  ->  b  =  ( F `  w ) )
10 fveq2 5690 . . . . . . . . 9  |-  ( w  =  W  ->  ( F `  w )  =  ( F `  W ) )
11 sbcie2s.b . . . . . . . . 9  |-  B  =  ( F `  W
)
1210, 11syl6eqr 2492 . . . . . . . 8  |-  ( w  =  W  ->  ( F `  w )  =  B )
1312adantr 465 . . . . . . 7  |-  ( ( w  =  W  /\  ( a  =  ( E `  w )  /\  b  =  ( F `  w ) ) )  ->  ( F `  w )  =  B )
149, 13eqtrd 2474 . . . . . 6  |-  ( ( w  =  W  /\  ( a  =  ( E `  w )  /\  b  =  ( F `  w ) ) )  ->  b  =  B )
158, 14jca 532 . . . . 5  |-  ( ( w  =  W  /\  ( a  =  ( E `  w )  /\  b  =  ( F `  w ) ) )  ->  (
a  =  A  /\  b  =  B )
)
16 sbcie2s.1 . . . . 5  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ph  <->  ps )
)
1715, 16syl 16 . . . 4  |-  ( ( w  =  W  /\  ( a  =  ( E `  w )  /\  b  =  ( F `  w ) ) )  ->  ( ph 
<->  ps ) )
1817bicomd 201 . . 3  |-  ( ( w  =  W  /\  ( a  =  ( E `  w )  /\  b  =  ( F `  w ) ) )  ->  ( ps 
<-> 
ph ) )
1918ex 434 . 2  |-  ( w  =  W  ->  (
( a  =  ( E `  w )  /\  b  =  ( F `  w ) )  ->  ( ps  <->  ph ) ) )
201, 2, 19sbc2iedv 3262 1  |-  ( w  =  W  ->  ( [. ( E `  w
)  /  a ]. [. ( F `  w
)  /  b ]. ps 
<-> 
ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   [.wsbc 3185   ` cfv 5417
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 4420
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-iota 5380  df-fv 5425
This theorem is referenced by:  istrkgc  22916  istrkgb  22917  istrkge  22919  istrkgl  22920
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