Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  onfrALTlem1VD Unicode version

Theorem onfrALTlem1VD 28711
Description: Virtual deduction proof of onfrALTlem1 28345. 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 28345 is onfrALTlem1VD 28711 without virtual deductions and was automatically derived from onfrALTlem1VD 28711.
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 28423 . . . . 5  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  ->.  ( x  e.  a  /\  (
a  i^i  x )  =  (/) ) ).
2 19.8a 1758 . . . . 5  |-  ( ( x  e.  a  /\  ( a  i^i  x
)  =  (/) )  ->  E. x ( x  e.  a  /\  ( a  i^i  x )  =  (/) ) )
31, 2e2 28441 . . . 4  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  ->.  E. x
( x  e.  a  /\  ( a  i^i  x )  =  (/) ) ).
4 cbvexsv 28344 . . . . 5  |-  ( E. x ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  <->  E. y [ y  /  x ] ( x  e.  a  /\  ( a  i^i  x
)  =  (/) ) )
54biimpi 187 . . . 4  |-  ( E. x ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  ->  E. y [ y  /  x ] ( x  e.  a  /\  ( a  i^i  x )  =  (/) ) )
63, 5e2 28441 . . 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 3125 . . . . . 6  |-  ( [ y  /  x ]
( x  e.  a  /\  ( a  i^i  x )  =  (/) ) 
<-> 
[. y  /  x ]. ( x  e.  a  /\  ( a  i^i  x )  =  (/) ) )
8 onfrALTlem4 28340 . . . . . 6  |-  ( [. y  /  x ]. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  <->  ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
97, 8bitri 241 . . . . 5  |-  ( [ y  /  x ]
( x  e.  a  /\  ( a  i^i  x )  =  (/) ) 
<->  ( y  e.  a  /\  ( a  i^i  y )  =  (/) ) )
109ax-gen 1552 . . . 4  |-  A. y
( [ y  /  x ] ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  <->  ( y  e.  a  /\  ( a  i^i  y )  =  (/) ) )
11 exbi 1588 . . . 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_ 28593 . . 3  |-  ( E. y [ y  /  x ] ( x  e.  a  /\  ( a  i^i  x )  =  (/) )  <->  E. y ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
136, 12e2bi 28442 . 2  |-  (. (
a  C_  On  /\  a  =/=  (/) ) ,. (
x  e.  a  /\  ( a  i^i  x
)  =  (/) )  ->.  E. y
( y  e.  a  /\  ( a  i^i  y )  =  (/) ) ).
14 df-rex 2672 . 2  |-  ( E. y  e.  a  ( a  i^i  y )  =  (/)  <->  E. y ( y  e.  a  /\  (
a  i^i  y )  =  (/) ) )
1513, 14e2bir 28443 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 177    /\ wa 359   A.wal 1546   E.wex 1547    = wceq 1649   [wsb 1655    e. wcel 1721    =/= wne 2567   E.wrex 2667   [.wsbc 3121    i^i cin 3279    C_ wss 3280   (/)c0 3588   Oncon0 4541   (.wvd2 28378
This theorem is referenced by:  onfrALTVD  28712
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-in 3287  df-vd2 28379
  Copyright terms: Public domain W3C validator