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Theorem onfrALTlem4VD 37346
Description: Virtual deduction proof of onfrALTlem4 36979. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. onfrALTlem4 36979 is onfrALTlem4VD 37346 without virtual deductions and was automatically derived from onfrALTlem4VD 37346.
1::  |-  y  e.  _V
2:1:  |-  ( [. y  /  x ]. ( a  i^i  x )  =  (/)  <->  [_  y  /  x ]_ ( a  i^i  x )  =  [_ y  /  x ]_ (/) )
3:1:  |-  [_ y  /  x ]_ ( a  i^i  x )  =  ( [_ y  /  x ]_  a  i^i  [_ y  /  x ]_ x )
4:1:  |-  [_ y  /  x ]_ a  =  a
5:1:  |-  [_ y  /  x ]_ x  =  y
6:4,5:  |-  ( [_ y  /  x ]_ a  i^i  [_ y  /  x ]_ x )  =  (  a  i^i  y )
7:3,6:  |-  [_ y  /  x ]_ ( a  i^i  x )  =  ( a  i^i  y )
8:1:  |-  [_ y  /  x ]_ (/)  =  (/)
9:7,8:  |-  ( [_ y  /  x ]_ ( a  i^i  x )  =  [_ y  /  x ]_  (/)  <->  ( a  i^i  y )  =  (/) )
10:2,9:  |-  ( [. y  /  x ]. ( a  i^i  x )  =  (/)  <->  ( a  i^i  y )  =  (/) )
11:1:  |-  ( [. y  /  x ]. x  e.  a  <->  y  e.  a )
12:11,10:  |-  ( ( [. y  /  x ]. x  e.  a  /\  [. y  /  x ]. (  a  i^i  x )  =  (/) )  <->  ( y  e.  a  /\  ( a  i^i  y )  =  (/) ) )
13:1:  |-  ( [. y  /  x ]. ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  <->  ( [. y  /  x ]. x  e.  a  /\  [. y  /  x ]. ( a  i^i  x )  =  (/) ) )
qed:13,12:  |-  ( [. y  /  x ]. ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  <->  ( y  e.  a  /\  ( a  i^i  y )  =  (/) ) )
(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
onfrALTlem4VD  |-  ( [. y  /  x ]. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  <->  ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
Distinct variable group:    x, a

Proof of Theorem onfrALTlem4VD
StepHypRef Expression
1 vex 3034 . . 3  |-  y  e. 
_V
2 sbcangOLD 36960 . . 3  |-  ( y  e.  _V  ->  ( [. y  /  x ]. ( x  e.  a  /\  ( a  i^i  x )  =  (/) ) 
<->  ( [. y  /  x ]. x  e.  a  /\  [. y  /  x ]. ( a  i^i  x )  =  (/) ) ) )
31, 2e0a 37222 . 2  |-  ( [. y  /  x ]. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  <->  ( [. y  /  x ]. x  e.  a  /\  [. y  /  x ]. ( a  i^i  x )  =  (/) ) )
4 sbcel1gvOLD 37318 . . . 4  |-  ( y  e.  _V  ->  ( [. y  /  x ]. x  e.  a  <->  y  e.  a ) )
51, 4e0a 37222 . . 3  |-  ( [. y  /  x ]. x  e.  a  <->  y  e.  a )
6 sbceqg 3777 . . . . 5  |-  ( y  e.  _V  ->  ( [. y  /  x ]. ( a  i^i  x
)  =  (/)  <->  [_ y  /  x ]_ ( a  i^i  x )  =  [_ y  /  x ]_ (/) ) )
71, 6e0a 37222 . . . 4  |-  ( [. y  /  x ]. (
a  i^i  x )  =  (/)  <->  [_ y  /  x ]_ ( a  i^i  x
)  =  [_ y  /  x ]_ (/) )
8 csbingOLD 37278 . . . . . . 7  |-  ( y  e.  _V  ->  [_ y  /  x ]_ ( a  i^i  x )  =  ( [_ y  /  x ]_ a  i^i  [_ y  /  x ]_ x ) )
91, 8e0a 37222 . . . . . 6  |-  [_ y  /  x ]_ ( a  i^i  x )  =  ( [_ y  /  x ]_ a  i^i  [_ y  /  x ]_ x )
10 csbconstg 3362 . . . . . . . 8  |-  ( y  e.  _V  ->  [_ y  /  x ]_ a  =  a )
111, 10e0a 37222 . . . . . . 7  |-  [_ y  /  x ]_ a  =  a
12 csbvarg 3796 . . . . . . . 8  |-  ( y  e.  _V  ->  [_ y  /  x ]_ x  =  y )
131, 12e0a 37222 . . . . . . 7  |-  [_ y  /  x ]_ x  =  y
1411, 13ineq12i 3623 . . . . . 6  |-  ( [_ y  /  x ]_ a  i^i  [_ y  /  x ]_ x )  =  ( a  i^i  y )
159, 14eqtri 2493 . . . . 5  |-  [_ y  /  x ]_ ( a  i^i  x )  =  ( a  i^i  y
)
16 csbconstg 3362 . . . . . 6  |-  ( y  e.  _V  ->  [_ y  /  x ]_ (/)  =  (/) )
171, 16e0a 37222 . . . . 5  |-  [_ y  /  x ]_ (/)  =  (/)
1815, 17eqeq12i 2485 . . . 4  |-  ( [_ y  /  x ]_ (
a  i^i  x )  =  [_ y  /  x ]_ (/)  <->  ( a  i^i  y )  =  (/) )
197, 18bitri 257 . . 3  |-  ( [. y  /  x ]. (
a  i^i  x )  =  (/)  <->  ( a  i^i  y )  =  (/) )
205, 19anbi12i 711 . 2  |-  ( (
[. y  /  x ]. x  e.  a  /\  [. y  /  x ]. ( a  i^i  x
)  =  (/) )  <->  ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
213, 20bitri 257 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 189    /\ wa 376    = wceq 1452    e. wcel 1904   _Vcvv 3031   [.wsbc 3255   [_csb 3349    i^i cin 3389   (/)c0 3722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-in 3397
This theorem is referenced by: (None)
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