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Theorem onfrALTlem4 28607
Description: Lemma for onfrALT 28613. (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 3046 . 2  |-  ( [. y  /  x ]. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  <->  ( [. y  /  x ]. x  e.  a  /\  [. y  /  x ]. ( a  i^i  x )  =  (/) ) )
2 vex 2804 . . . 4  |-  y  e. 
_V
3 sbcel1gv 3063 . . . 4  |-  ( y  e.  _V  ->  ( [. y  /  x ]. x  e.  a  <->  y  e.  a ) )
42, 3ax-mp 8 . . 3  |-  ( [. y  /  x ]. x  e.  a  <->  y  e.  a )
5 sbceqg 3110 . . . . 5  |-  ( y  e.  _V  ->  ( [. y  /  x ]. ( a  i^i  x
)  =  (/)  <->  [_ y  /  x ]_ ( a  i^i  x )  =  [_ y  /  x ]_ (/) ) )
62, 5ax-mp 8 . . . 4  |-  ( [. y  /  x ]. (
a  i^i  x )  =  (/)  <->  [_ y  /  x ]_ ( a  i^i  x
)  =  [_ y  /  x ]_ (/) )
7 csbing 3389 . . . . . . 7  |-  ( y  e.  _V  ->  [_ y  /  x ]_ ( a  i^i  x )  =  ( [_ y  /  x ]_ a  i^i  [_ y  /  x ]_ x ) )
82, 7ax-mp 8 . . . . . 6  |-  [_ y  /  x ]_ ( a  i^i  x )  =  ( [_ y  /  x ]_ a  i^i  [_ y  /  x ]_ x )
9 csbconstg 3108 . . . . . . . 8  |-  ( y  e.  _V  ->  [_ y  /  x ]_ a  =  a )
102, 9ax-mp 8 . . . . . . 7  |-  [_ y  /  x ]_ a  =  a
11 csbvarg 3121 . . . . . . . 8  |-  ( y  e.  _V  ->  [_ y  /  x ]_ x  =  y )
122, 11ax-mp 8 . . . . . . 7  |-  [_ y  /  x ]_ x  =  y
1310, 12ineq12i 3381 . . . . . 6  |-  ( [_ y  /  x ]_ a  i^i  [_ y  /  x ]_ x )  =  ( a  i^i  y )
148, 13eqtri 2316 . . . . 5  |-  [_ y  /  x ]_ ( a  i^i  x )  =  ( a  i^i  y
)
15 csbconstg 3108 . . . . . 6  |-  ( y  e.  _V  ->  [_ y  /  x ]_ (/)  =  (/) )
162, 15ax-mp 8 . . . . 5  |-  [_ y  /  x ]_ (/)  =  (/)
1714, 16eqeq12i 2309 . . . 4  |-  ( [_ y  /  x ]_ (
a  i^i  x )  =  [_ y  /  x ]_ (/)  <->  ( a  i^i  y )  =  (/) )
186, 17bitri 240 . . 3  |-  ( [. y  /  x ]. (
a  i^i  x )  =  (/)  <->  ( a  i^i  y )  =  (/) )
194, 18anbi12i 678 . 2  |-  ( (
[. y  /  x ]. x  e.  a  /\  [. y  /  x ]. ( a  i^i  x
)  =  (/) )  <->  ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
201, 19bitri 240 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   [.wsbc 3004   [_csb 3094    i^i cin 3164   (/)c0 3468
This theorem is referenced by:  onfrALTlem1  28612  onfrALTlem1VD  28982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-in 3172
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