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Theorem sbcie2s 14688
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 5882 . 2  |-  ( E `
 w )  e. 
_V
2 fvex 5882 . 2  |-  ( F `
 w )  e. 
_V
3 simprl 756 . . . . . 6  |-  ( ( w  =  W  /\  ( a  =  ( E `  w )  /\  b  =  ( F `  w ) ) )  ->  a  =  ( E `  w ) )
4 fveq2 5872 . . . . . . . 8  |-  ( w  =  W  ->  ( E `  w )  =  ( E `  W ) )
5 sbcie2s.a . . . . . . . 8  |-  A  =  ( E `  W
)
64, 5syl6eqr 2516 . . . . . . 7  |-  ( w  =  W  ->  ( E `  w )  =  A )
76adantr 465 . . . . . 6  |-  ( ( w  =  W  /\  ( a  =  ( E `  w )  /\  b  =  ( F `  w ) ) )  ->  ( E `  w )  =  A )
83, 7eqtrd 2498 . . . . 5  |-  ( ( w  =  W  /\  ( a  =  ( E `  w )  /\  b  =  ( F `  w ) ) )  ->  a  =  A )
9 simprr 757 . . . . . 6  |-  ( ( w  =  W  /\  ( a  =  ( E `  w )  /\  b  =  ( F `  w ) ) )  ->  b  =  ( F `  w ) )
10 fveq2 5872 . . . . . . . 8  |-  ( w  =  W  ->  ( F `  w )  =  ( F `  W ) )
11 sbcie2s.b . . . . . . . 8  |-  B  =  ( F `  W
)
1210, 11syl6eqr 2516 . . . . . . 7  |-  ( w  =  W  ->  ( F `  w )  =  B )
1312adantr 465 . . . . . 6  |-  ( ( w  =  W  /\  ( a  =  ( E `  w )  /\  b  =  ( F `  w ) ) )  ->  ( F `  w )  =  B )
149, 13eqtrd 2498 . . . . 5  |-  ( ( w  =  W  /\  ( a  =  ( E `  w )  /\  b  =  ( F `  w ) ) )  ->  b  =  B )
15 sbcie2s.1 . . . . 5  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ph  <->  ps )
)
168, 14, 15syl2anc 661 . . . 4  |-  ( ( w  =  W  /\  ( a  =  ( E `  w )  /\  b  =  ( F `  w ) ) )  ->  ( ph 
<->  ps ) )
1716bicomd 201 . . 3  |-  ( ( w  =  W  /\  ( a  =  ( E `  w )  /\  b  =  ( F `  w ) ) )  ->  ( ps 
<-> 
ph ) )
1817ex 434 . 2  |-  ( w  =  W  ->  (
( a  =  ( E `  w )  /\  b  =  ( F `  w ) )  ->  ( ps  <->  ph ) ) )
191, 2, 18sbc2iedv 3402 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 1395   [.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:  istrkgc  23976  istrkgb  23977  istrkge  23979  istrkgl  23980  ishpg  24253
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