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Theorem nabbiOLD 2760
Description: Obsolete proof of nabbi 2759 as of 25-Nov-2019. (Contributed by AV, 7-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
nabbiOLD  |-  ( E. x ( ph  <->  -.  ps )  <->  { x  |  ph }  =/=  { x  |  ps } )

Proof of Theorem nabbiOLD
StepHypRef Expression
1 df-ne 2621 . . 3  |-  ( { x  |  ph }  =/=  { x  |  ps } 
<->  -.  { x  | 
ph }  =  {
x  |  ps }
)
2 abbi 2554 . . . . . 6  |-  ( A. x ( ph  <->  ps )  <->  { x  |  ph }  =  { x  |  ps } )
32bicomi 206 . . . . 5  |-  ( { x  |  ph }  =  { x  |  ps } 
<-> 
A. x ( ph  <->  ps ) )
43notbii 298 . . . 4  |-  ( -. 
{ x  |  ph }  =  { x  |  ps }  <->  -.  A. x
( ph  <->  ps ) )
5 exnal 1696 . . . . . 6  |-  ( E. x  -.  ( ph  <->  ps )  <->  -.  A. x
( ph  <->  ps ) )
65bicomi 206 . . . . 5  |-  ( -. 
A. x ( ph  <->  ps )  <->  E. x  -.  ( ph 
<->  ps ) )
7 xor3 359 . . . . . 6  |-  ( -.  ( ph  <->  ps )  <->  (
ph 
<->  -.  ps ) )
87exbii 1713 . . . . 5  |-  ( E. x  -.  ( ph  <->  ps )  <->  E. x ( ph  <->  -. 
ps ) )
96, 8bitri 253 . . . 4  |-  ( -. 
A. x ( ph  <->  ps )  <->  E. x ( ph  <->  -. 
ps ) )
104, 9bitri 253 . . 3  |-  ( -. 
{ x  |  ph }  =  { x  |  ps }  <->  E. x
( ph  <->  -.  ps )
)
111, 10bitri 253 . 2  |-  ( { x  |  ph }  =/=  { x  |  ps } 
<->  E. x ( ph  <->  -. 
ps ) )
1211bicomi 206 1  |-  ( E. x ( ph  <->  -.  ps )  <->  { x  |  ph }  =/=  { x  |  ps } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 188   A.wal 1436    = wceq 1438   E.wex 1660   {cab 2408    =/= wne 2619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-ne 2621
This theorem is referenced by: (None)
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