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Theorem onfrALTlem4VD 28173
Description: Virtual deduction proof of onfrALTlem4 27802. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. onfrALTlem4 27802 is onfrALTlem4VD 28173 without virtual deductions and was automatically derived from onfrALTlem4VD 28173.
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 2825 . . 3  |-  y  e. 
_V
2 sbcang 3068 . . 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, 2e0_ 28056 . 2  |-  ( [. y  /  x ]. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  <->  ( [. y  /  x ]. x  e.  a  /\  [. y  /  x ]. ( a  i^i  x )  =  (/) ) )
4 sbcel1gv 3084 . . . 4  |-  ( y  e.  _V  ->  ( [. y  /  x ]. x  e.  a  <->  y  e.  a ) )
51, 4e0_ 28056 . . 3  |-  ( [. y  /  x ]. x  e.  a  <->  y  e.  a )
6 sbceqg 3131 . . . . 5  |-  ( y  e.  _V  ->  ( [. y  /  x ]. ( a  i^i  x
)  =  (/)  <->  [_ y  /  x ]_ ( a  i^i  x )  =  [_ y  /  x ]_ (/) ) )
71, 6e0_ 28056 . . . 4  |-  ( [. y  /  x ]. (
a  i^i  x )  =  (/)  <->  [_ y  /  x ]_ ( a  i^i  x
)  =  [_ y  /  x ]_ (/) )
8 csbing 3410 . . . . . . 7  |-  ( y  e.  _V  ->  [_ y  /  x ]_ ( a  i^i  x )  =  ( [_ y  /  x ]_ a  i^i  [_ y  /  x ]_ x ) )
91, 8e0_ 28056 . . . . . 6  |-  [_ y  /  x ]_ ( a  i^i  x )  =  ( [_ y  /  x ]_ a  i^i  [_ y  /  x ]_ x )
10 csbconstg 3129 . . . . . . . 8  |-  ( y  e.  _V  ->  [_ y  /  x ]_ a  =  a )
111, 10e0_ 28056 . . . . . . 7  |-  [_ y  /  x ]_ a  =  a
12 csbvarg 3142 . . . . . . . 8  |-  ( y  e.  _V  ->  [_ y  /  x ]_ x  =  y )
131, 12e0_ 28056 . . . . . . 7  |-  [_ y  /  x ]_ x  =  y
1411, 13ineq12i 3402 . . . . . 6  |-  ( [_ y  /  x ]_ a  i^i  [_ y  /  x ]_ x )  =  ( a  i^i  y )
159, 14eqtri 2336 . . . . 5  |-  [_ y  /  x ]_ ( a  i^i  x )  =  ( a  i^i  y
)
16 csbconstg 3129 . . . . . 6  |-  ( y  e.  _V  ->  [_ y  /  x ]_ (/)  =  (/) )
171, 16e0_ 28056 . . . . 5  |-  [_ y  /  x ]_ (/)  =  (/)
1815, 17eqeq12i 2329 . . . 4  |-  ( [_ y  /  x ]_ (
a  i^i  x )  =  [_ y  /  x ]_ (/)  <->  ( a  i^i  y )  =  (/) )
197, 18bitri 240 . . 3  |-  ( [. y  /  x ]. (
a  i^i  x )  =  (/)  <->  ( a  i^i  y )  =  (/) )
205, 19anbi12i 678 . 2  |-  ( (
[. y  /  x ]. x  e.  a  /\  [. y  /  x ]. ( a  i^i  x
)  =  (/) )  <->  ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
213, 20bitri 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 1633    e. wcel 1701   _Vcvv 2822   [.wsbc 3025   [_csb 3115    i^i cin 3185   (/)c0 3489
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-in 3193
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