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Theorem isfildlem 17811
Description: Lemma for isfild 17812. (Contributed by Mario Carneiro, 1-Dec-2013.)
Hypotheses
Ref Expression
isfild.1  |-  ( ph  ->  ( x  e.  F  <->  ( x  C_  A  /\  ps ) ) )
isfild.2  |-  ( ph  ->  A  e.  _V )
Assertion
Ref Expression
isfildlem  |-  ( ph  ->  ( B  e.  F  <->  ( B  C_  A  /\  [. B  /  x ]. ps ) ) )
Distinct variable groups:    x, A    x, F    ph, x
Allowed substitution hints:    ps( x)    B( x)

Proof of Theorem isfildlem
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 elex 2908 . . 3  |-  ( B  e.  F  ->  B  e.  _V )
21a1i 11 . 2  |-  ( ph  ->  ( B  e.  F  ->  B  e.  _V )
)
3 isfild.2 . . . 4  |-  ( ph  ->  A  e.  _V )
4 ssexg 4291 . . . . 5  |-  ( ( B  C_  A  /\  A  e.  _V )  ->  B  e.  _V )
54expcom 425 . . . 4  |-  ( A  e.  _V  ->  ( B  C_  A  ->  B  e.  _V ) )
63, 5syl 16 . . 3  |-  ( ph  ->  ( B  C_  A  ->  B  e.  _V )
)
76adantrd 455 . 2  |-  ( ph  ->  ( ( B  C_  A  /\  [. B  /  x ]. ps )  ->  B  e.  _V )
)
8 eleq1 2448 . . . . . 6  |-  ( y  =  B  ->  (
y  e.  F  <->  B  e.  F ) )
9 sseq1 3313 . . . . . . 7  |-  ( y  =  B  ->  (
y  C_  A  <->  B  C_  A
) )
10 dfsbcq 3107 . . . . . . 7  |-  ( y  =  B  ->  ( [. y  /  x ]. ps  <->  [. B  /  x ]. ps ) )
119, 10anbi12d 692 . . . . . 6  |-  ( y  =  B  ->  (
( y  C_  A  /\  [. y  /  x ]. ps )  <->  ( B  C_  A  /\  [. B  /  x ]. ps )
) )
128, 11bibi12d 313 . . . . 5  |-  ( y  =  B  ->  (
( y  e.  F  <->  ( y  C_  A  /\  [. y  /  x ]. ps ) )  <->  ( B  e.  F  <->  ( B  C_  A  /\  [. B  /  x ]. ps ) ) ) )
1312imbi2d 308 . . . 4  |-  ( y  =  B  ->  (
( ph  ->  ( y  e.  F  <->  ( y  C_  A  /\  [. y  /  x ]. ps )
) )  <->  ( ph  ->  ( B  e.  F  <->  ( B  C_  A  /\  [. B  /  x ]. ps ) ) ) ) )
14 nfv 1626 . . . . . 6  |-  F/ x ph
15 nfv 1626 . . . . . . 7  |-  F/ x  y  e.  F
16 nfv 1626 . . . . . . . 8  |-  F/ x  y  C_  A
17 nfsbc1v 3124 . . . . . . . 8  |-  F/ x [. y  /  x ]. ps
1816, 17nfan 1836 . . . . . . 7  |-  F/ x
( y  C_  A  /\  [. y  /  x ]. ps )
1915, 18nfbi 1846 . . . . . 6  |-  F/ x
( y  e.  F  <->  ( y  C_  A  /\  [. y  /  x ]. ps ) )
2014, 19nfim 1822 . . . . 5  |-  F/ x
( ph  ->  ( y  e.  F  <->  ( y  C_  A  /\  [. y  /  x ]. ps )
) )
21 eleq1 2448 . . . . . . 7  |-  ( x  =  y  ->  (
x  e.  F  <->  y  e.  F ) )
22 sseq1 3313 . . . . . . . 8  |-  ( x  =  y  ->  (
x  C_  A  <->  y  C_  A ) )
23 sbceq1a 3115 . . . . . . . 8  |-  ( x  =  y  ->  ( ps 
<-> 
[. y  /  x ]. ps ) )
2422, 23anbi12d 692 . . . . . . 7  |-  ( x  =  y  ->  (
( x  C_  A  /\  ps )  <->  ( y  C_  A  /\  [. y  /  x ]. ps )
) )
2521, 24bibi12d 313 . . . . . 6  |-  ( x  =  y  ->  (
( x  e.  F  <->  ( x  C_  A  /\  ps ) )  <->  ( y  e.  F  <->  ( y  C_  A  /\  [. y  /  x ]. ps ) ) ) )
2625imbi2d 308 . . . . 5  |-  ( x  =  y  ->  (
( ph  ->  ( x  e.  F  <->  ( x  C_  A  /\  ps )
) )  <->  ( ph  ->  ( y  e.  F  <->  ( y  C_  A  /\  [. y  /  x ]. ps ) ) ) ) )
27 isfild.1 . . . . 5  |-  ( ph  ->  ( x  e.  F  <->  ( x  C_  A  /\  ps ) ) )
2820, 26, 27chvar 2023 . . . 4  |-  ( ph  ->  ( y  e.  F  <->  ( y  C_  A  /\  [. y  /  x ]. ps ) ) )
2913, 28vtoclg 2955 . . 3  |-  ( B  e.  _V  ->  ( ph  ->  ( B  e.  F  <->  ( B  C_  A  /\  [. B  /  x ]. ps ) ) ) )
3029com12 29 . 2  |-  ( ph  ->  ( B  e.  _V  ->  ( B  e.  F  <->  ( B  C_  A  /\  [. B  /  x ]. ps ) ) ) )
312, 7, 30pm5.21ndd 344 1  |-  ( ph  ->  ( B  e.  F  <->  ( B  C_  A  /\  [. B  /  x ]. ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2900   [.wsbc 3105    C_ wss 3264
This theorem is referenced by:  isfild  17812
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-v 2902  df-sbc 3106  df-in 3271  df-ss 3278
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