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Theorem onfrALTlem4 28801
Description: Lemma for onfrALT 28807. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
onfrALTlem4  |-  ( [. y  /  x ]. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  <->  ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
Distinct variable group:    x, a

Proof of Theorem onfrALTlem4
StepHypRef Expression
1 sbcan 3212 . 2  |-  ( [. y  /  x ]. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  <->  ( [. y  /  x ]. x  e.  a  /\  [. y  /  x ]. ( a  i^i  x )  =  (/) ) )
2 vex 2968 . . . 4  |-  y  e. 
_V
3 sbcel1gvOLD 3234 . . . 4  |-  ( y  e.  _V  ->  ( [. y  /  x ]. x  e.  a  <->  y  e.  a ) )
42, 3ax-mp 5 . . 3  |-  ( [. y  /  x ]. x  e.  a  <->  y  e.  a )
5 sbceqg 3655 . . . . 5  |-  ( y  e.  _V  ->  ( [. y  /  x ]. ( a  i^i  x
)  =  (/)  <->  [_ y  /  x ]_ ( a  i^i  x )  =  [_ y  /  x ]_ (/) ) )
62, 5ax-mp 5 . . . 4  |-  ( [. y  /  x ]. (
a  i^i  x )  =  (/)  <->  [_ y  /  x ]_ ( a  i^i  x
)  =  [_ y  /  x ]_ (/) )
7 csbingOLD 3690 . . . . . . 7  |-  ( y  e.  _V  ->  [_ y  /  x ]_ ( a  i^i  x )  =  ( [_ y  /  x ]_ a  i^i  [_ y  /  x ]_ x ) )
82, 7ax-mp 5 . . . . . 6  |-  [_ y  /  x ]_ ( a  i^i  x )  =  ( [_ y  /  x ]_ a  i^i  [_ y  /  x ]_ x )
9 csbconstg 3282 . . . . . . . 8  |-  ( y  e.  _V  ->  [_ y  /  x ]_ a  =  a )
102, 9ax-mp 5 . . . . . . 7  |-  [_ y  /  x ]_ a  =  a
11 csbvarg 3680 . . . . . . . 8  |-  ( y  e.  _V  ->  [_ y  /  x ]_ x  =  y )
122, 11ax-mp 5 . . . . . . 7  |-  [_ y  /  x ]_ x  =  y
1310, 12ineq12i 3529 . . . . . 6  |-  ( [_ y  /  x ]_ a  i^i  [_ y  /  x ]_ x )  =  ( a  i^i  y )
148, 13eqtri 2463 . . . . 5  |-  [_ y  /  x ]_ ( a  i^i  x )  =  ( a  i^i  y
)
15 csbconstg 3282 . . . . . 6  |-  ( y  e.  _V  ->  [_ y  /  x ]_ (/)  =  (/) )
162, 15ax-mp 5 . . . . 5  |-  [_ y  /  x ]_ (/)  =  (/)
1714, 16eqeq12i 2456 . . . 4  |-  ( [_ y  /  x ]_ (
a  i^i  x )  =  [_ y  /  x ]_ (/)  <->  ( a  i^i  y )  =  (/) )
186, 17bitri 242 . . 3  |-  ( [. y  /  x ]. (
a  i^i  x )  =  (/)  <->  ( a  i^i  y )  =  (/) )
194, 18anbi12i 680 . 2  |-  ( (
[. y  /  x ]. x  e.  a  /\  [. y  /  x ]. ( a  i^i  x
)  =  (/) )  <->  ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
201, 19bitri 242 1  |-  ( [. y  /  x ]. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  <->  ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1654    e. wcel 1728   _Vcvv 2965   [.wsbc 3170   [_csb 3270    i^i cin 3308   (/)c0 3616
This theorem is referenced by:  onfrALTlem1  28806  onfrALTlem1VD  29176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-in 3316
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