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Theorem onfrALTlem1VD 28666
Description: Virtual deduction proof of onfrALTlem1 28313. 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. onfrALTlem1 28313 is onfrALTlem1VD 28666 without virtual deductions and was automatically derived from onfrALTlem1VD 28666.
1::  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  ->.  ( x  e.  a  /\  ( a  i^i  x )  =  (/) ) ).
2:1:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  ->.  E. x ( x  e.  a  /\  ( a  i^i  x )  =  (/) ) ).
3:2:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  ->.  E. y [ y  /  x ] ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  ).
4::  |-  ( [ y  /  x ] ( x  e.  a  /\  ( a  i^i  x )  =  (/)  )  <->  ( y  e.  a  /\  ( a  i^i  y )  =  (/) ) )
5:4:  |-  A. y ( [ y  /  x ] ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  <->  ( y  e.  a  /\  ( a  i^i  y )  =  (/) ) )
6:5:  |-  ( E. y [ y  /  x ] ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  <->  E. y ( y  e.  a  /\  ( a  i^i  y )  =  (/) ) )
7:3,6:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  ->.  E. y ( y  e.  a  /\  ( a  i^i  y )  =  (/) ) ).
8::  |-  ( E. y  e.  a ( a  i^i  y )  =  (/)  <->  E. y (  y  e.  a  /\  ( a  i^i  y )  =  (/) ) )
qed:7,8:  |-  (. ( a  C_  On  /\  a  =/=  (/) ) ,. ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  ->.  E. 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
onfrALTlem1VD  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  ->.  E. y  e.  a  ( a  i^i  y )  =  (/) ).
Distinct variable group:    x, a, y

Proof of Theorem onfrALTlem1VD
StepHypRef Expression
1 idn2 28385 . . . . 5  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  ->.  ( x  e.  a  /\  (
a  i^i  x )  =  (/) ) ).
2 19.8a 1718 . . . . 5  |-  ( ( x  e.  a  /\  ( a  i^i  x
)  =  (/) )  ->  E. x ( x  e.  a  /\  ( a  i^i  x )  =  (/) ) )
31, 2e2 28403 . . . 4  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  ->.  E. x
( x  e.  a  /\  ( a  i^i  x )  =  (/) ) ).
4 cbvexsv 28312 . . . . 5  |-  ( E. x ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  <->  E. y [ y  /  x ] ( x  e.  a  /\  ( a  i^i  x
)  =  (/) ) )
54biimpi 186 . . . 4  |-  ( E. x ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  ->  E. y [ y  /  x ] ( x  e.  a  /\  ( a  i^i  x )  =  (/) ) )
63, 5e2 28403 . . 3  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  ->.  E. y [ y  /  x ] ( x  e.  a  /\  ( a  i^i  x )  =  (/) ) ).
7 sbsbc 2995 . . . . . 6  |-  ( [ y  /  x ]
( x  e.  a  /\  ( a  i^i  x )  =  (/) ) 
<-> 
[. y  /  x ]. ( x  e.  a  /\  ( a  i^i  x )  =  (/) ) )
8 onfrALTlem4 28308 . . . . . 6  |-  ( [. y  /  x ]. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  <->  ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
97, 8bitri 240 . . . . 5  |-  ( [ y  /  x ]
( x  e.  a  /\  ( a  i^i  x )  =  (/) ) 
<->  ( y  e.  a  /\  ( a  i^i  y )  =  (/) ) )
109ax-gen 1533 . . . 4  |-  A. y
( [ y  /  x ] ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  <->  ( y  e.  a  /\  ( a  i^i  y )  =  (/) ) )
11 exbi 1568 . . . 4  |-  ( A. y ( [ y  /  x ] ( x  e.  a  /\  ( a  i^i  x
)  =  (/) )  <->  ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )  -> 
( E. y [ y  /  x ]
( x  e.  a  /\  ( a  i^i  x )  =  (/) ) 
<->  E. y ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) ) )
1210, 11e0_ 28547 . . 3  |-  ( E. y [ y  /  x ] ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  <->  E. y ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
136, 12e2bi 28404 . 2  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  ->.  E. y
( y  e.  a  /\  ( a  i^i  y )  =  (/) ) ).
14 df-rex 2549 . 2  |-  ( E. y  e.  a  ( a  i^i  y )  =  (/)  <->  E. y ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
1513, 14e2bir 28405 1  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  ->.  E. y  e.  a  ( a  i^i  y )  =  (/) ).
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623   [wsb 1629    e. wcel 1684    =/= wne 2446   E.wrex 2544   [.wsbc 2991    i^i cin 3151    C_ wss 3152   (/)c0 3455   Oncon0 4392   (.wvd2 28346
This theorem is referenced by:  onfrALTVD  28667
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-in 3159  df-vd2 28347
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