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Theorem onfrALTlem4 36549
Description: Lemma for onfrALT 36555. (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 3348 . 2  |-  ( [. y  /  x ]. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  <->  ( [. y  /  x ]. x  e.  a  /\  [. y  /  x ]. ( a  i^i  x )  =  (/) ) )
2 sbcel1v 3364 . . 3  |-  ( [. y  /  x ]. x  e.  a  <->  y  e.  a )
3 vex 3090 . . . . 5  |-  y  e. 
_V
4 sbceqg 3807 . . . . 5  |-  ( y  e.  _V  ->  ( [. y  /  x ]. ( a  i^i  x
)  =  (/)  <->  [_ y  /  x ]_ ( a  i^i  x )  =  [_ y  /  x ]_ (/) ) )
53, 4ax-mp 5 . . . 4  |-  ( [. y  /  x ]. (
a  i^i  x )  =  (/)  <->  [_ y  /  x ]_ ( a  i^i  x
)  =  [_ y  /  x ]_ (/) )
6 csbin 3833 . . . . . 6  |-  [_ y  /  x ]_ ( a  i^i  x )  =  ( [_ y  /  x ]_ a  i^i  [_ y  /  x ]_ x )
7 csbconstg 3414 . . . . . . . 8  |-  ( y  e.  _V  ->  [_ y  /  x ]_ a  =  a )
83, 7ax-mp 5 . . . . . . 7  |-  [_ y  /  x ]_ a  =  a
9 csbvarg 3826 . . . . . . . 8  |-  ( y  e.  _V  ->  [_ y  /  x ]_ x  =  y )
103, 9ax-mp 5 . . . . . . 7  |-  [_ y  /  x ]_ x  =  y
118, 10ineq12i 3668 . . . . . 6  |-  ( [_ y  /  x ]_ a  i^i  [_ y  /  x ]_ x )  =  ( a  i^i  y )
126, 11eqtri 2458 . . . . 5  |-  [_ y  /  x ]_ ( a  i^i  x )  =  ( a  i^i  y
)
13 csb0 3805 . . . . 5  |-  [_ y  /  x ]_ (/)  =  (/)
1412, 13eqeq12i 2449 . . . 4  |-  ( [_ y  /  x ]_ (
a  i^i  x )  =  [_ y  /  x ]_ (/)  <->  ( a  i^i  y )  =  (/) )
155, 14bitri 252 . . 3  |-  ( [. y  /  x ]. (
a  i^i  x )  =  (/)  <->  ( a  i^i  y )  =  (/) )
162, 15anbi12i 701 . 2  |-  ( (
[. y  /  x ]. x  e.  a  /\  [. y  /  x ]. ( a  i^i  x
)  =  (/) )  <->  ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
171, 16bitri 252 1  |-  ( [. y  /  x ]. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  <->  ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087   [.wsbc 3305   [_csb 3401    i^i cin 3441   (/)c0 3767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-in 3449  df-ss 3456  df-nul 3768
This theorem is referenced by:  onfrALTlem1  36554  onfrALTlem1VD  36930
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