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Theorem nabbi 2758
Description: Not equivalent wff's correspond to not equal class abstractions. (Contributed by AV, 7-Apr-2019.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
Assertion
Ref Expression
nabbi  |-  ( E. x ( ph  <->  -.  ps )  <->  { x  |  ph }  =/=  { x  |  ps } )

Proof of Theorem nabbi
StepHypRef Expression
1 df-ne 2620 . 2  |-  ( { x  |  ph }  =/=  { x  |  ps } 
<->  -.  { x  | 
ph }  =  {
x  |  ps }
)
2 exnal 1695 . . . 4  |-  ( E. x  -.  ( ph  <->  ps )  <->  -.  A. x
( ph  <->  ps ) )
3 xor3 358 . . . . 5  |-  ( -.  ( ph  <->  ps )  <->  (
ph 
<->  -.  ps ) )
43exbii 1712 . . . 4  |-  ( E. x  -.  ( ph  <->  ps )  <->  E. x ( ph  <->  -. 
ps ) )
52, 4bitr3i 254 . . 3  |-  ( -. 
A. x ( ph  <->  ps )  <->  E. x ( ph  <->  -. 
ps ) )
6 abbi 2553 . . 3  |-  ( A. x ( ph  <->  ps )  <->  { x  |  ph }  =  { x  |  ps } )
75, 6xchnxbi 309 . 2  |-  ( -. 
{ x  |  ph }  =  { x  |  ps }  <->  E. x
( ph  <->  -.  ps )
)
81, 7bitr2i 253 1  |-  ( E. x ( ph  <->  -.  ps )  <->  { x  |  ph }  =/=  { x  |  ps } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187   A.wal 1435    = wceq 1437   E.wex 1659   {cab 2407    =/= wne 2618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-ne 2620
This theorem is referenced by:  suppvalbr  6925
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