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Theorem isfildlem 20872
Description: Lemma for isfild 20873. (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 3054 . . 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 4549 . . . . 5  |-  ( ( B  C_  A  /\  A  e.  _V )  ->  B  e.  _V )
54expcom 437 . . . 4  |-  ( A  e.  _V  ->  ( B  C_  A  ->  B  e.  _V ) )
63, 5syl 17 . . 3  |-  ( ph  ->  ( B  C_  A  ->  B  e.  _V )
)
76adantrd 470 . 2  |-  ( ph  ->  ( ( B  C_  A  /\  [. B  /  x ]. ps )  ->  B  e.  _V )
)
8 eleq1 2517 . . . . . 6  |-  ( y  =  B  ->  (
y  e.  F  <->  B  e.  F ) )
9 sseq1 3453 . . . . . . 7  |-  ( y  =  B  ->  (
y  C_  A  <->  B  C_  A
) )
10 dfsbcq 3269 . . . . . . 7  |-  ( y  =  B  ->  ( [. y  /  x ]. ps  <->  [. B  /  x ]. ps ) )
119, 10anbi12d 717 . . . . . 6  |-  ( y  =  B  ->  (
( y  C_  A  /\  [. y  /  x ]. ps )  <->  ( B  C_  A  /\  [. B  /  x ]. ps )
) )
128, 11bibi12d 323 . . . . 5  |-  ( y  =  B  ->  (
( y  e.  F  <->  ( y  C_  A  /\  [. y  /  x ]. ps ) )  <->  ( B  e.  F  <->  ( B  C_  A  /\  [. B  /  x ]. ps ) ) ) )
1312imbi2d 318 . . . 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 1761 . . . . . 6  |-  F/ x ph
15 nfv 1761 . . . . . . 7  |-  F/ x  y  e.  F
16 nfv 1761 . . . . . . . 8  |-  F/ x  y  C_  A
17 nfsbc1v 3287 . . . . . . . 8  |-  F/ x [. y  /  x ]. ps
1816, 17nfan 2011 . . . . . . 7  |-  F/ x
( y  C_  A  /\  [. y  /  x ]. ps )
1915, 18nfbi 2017 . . . . . 6  |-  F/ x
( y  e.  F  <->  ( y  C_  A  /\  [. y  /  x ]. ps ) )
2014, 19nfim 2003 . . . . 5  |-  F/ x
( ph  ->  ( y  e.  F  <->  ( y  C_  A  /\  [. y  /  x ]. ps )
) )
21 eleq1 2517 . . . . . . 7  |-  ( x  =  y  ->  (
x  e.  F  <->  y  e.  F ) )
22 sseq1 3453 . . . . . . . 8  |-  ( x  =  y  ->  (
x  C_  A  <->  y  C_  A ) )
23 sbceq1a 3278 . . . . . . . 8  |-  ( x  =  y  ->  ( ps 
<-> 
[. y  /  x ]. ps ) )
2422, 23anbi12d 717 . . . . . . 7  |-  ( x  =  y  ->  (
( x  C_  A  /\  ps )  <->  ( y  C_  A  /\  [. y  /  x ]. ps )
) )
2521, 24bibi12d 323 . . . . . 6  |-  ( x  =  y  ->  (
( x  e.  F  <->  ( x  C_  A  /\  ps ) )  <->  ( y  e.  F  <->  ( y  C_  A  /\  [. y  /  x ]. ps ) ) ) )
2625imbi2d 318 . . . . 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 2106 . . . 4  |-  ( ph  ->  ( y  e.  F  <->  ( y  C_  A  /\  [. y  /  x ]. ps ) ) )
2913, 28vtoclg 3107 . . 3  |-  ( B  e.  _V  ->  ( ph  ->  ( B  e.  F  <->  ( B  C_  A  /\  [. B  /  x ]. ps ) ) ) )
3029com12 32 . 2  |-  ( ph  ->  ( B  e.  _V  ->  ( B  e.  F  <->  ( B  C_  A  /\  [. B  /  x ]. ps ) ) ) )
312, 7, 30pm5.21ndd 356 1  |-  ( ph  ->  ( B  e.  F  <->  ( B  C_  A  /\  [. B  /  x ]. ps ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   _Vcvv 3045   [.wsbc 3267    C_ wss 3404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-sbc 3268  df-in 3411  df-ss 3418
This theorem is referenced by:  isfild  20873
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