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Theorem dtruALT2 4658
Description: Alternate proof of dtru 4608 using ax-pr 4653 instead of ax-pow 4595. (Contributed by Mario Carneiro, 31-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dtruALT2  |-  -.  A. x  x  =  y
Distinct variable group:    x, y

Proof of Theorem dtruALT2
StepHypRef Expression
1 0inp0 4589 . . . 4  |-  ( y  =  (/)  ->  -.  y  =  { (/) } )
2 snex 4655 . . . . 5  |-  { (/) }  e.  _V
3 eqeq2 2435 . . . . . 6  |-  ( x  =  { (/) }  ->  ( y  =  x  <->  y  =  { (/) } ) )
43notbid 295 . . . . 5  |-  ( x  =  { (/) }  ->  ( -.  y  =  x  <->  -.  y  =  { (/)
} ) )
52, 4spcev 3170 . . . 4  |-  ( -.  y  =  { (/) }  ->  E. x  -.  y  =  x )
61, 5syl 17 . . 3  |-  ( y  =  (/)  ->  E. x  -.  y  =  x
)
7 0ex 4549 . . . 4  |-  (/)  e.  _V
8 eqeq2 2435 . . . . 5  |-  ( x  =  (/)  ->  ( y  =  x  <->  y  =  (/) ) )
98notbid 295 . . . 4  |-  ( x  =  (/)  ->  ( -.  y  =  x  <->  -.  y  =  (/) ) )
107, 9spcev 3170 . . 3  |-  ( -.  y  =  (/)  ->  E. x  -.  y  =  x
)
116, 10pm2.61i 167 . 2  |-  E. x  -.  y  =  x
12 exnal 1695 . . 3  |-  ( E. x  -.  y  =  x  <->  -.  A. x  y  =  x )
13 eqcom 2429 . . . 4  |-  ( y  =  x  <->  x  =  y )
1413albii 1687 . . 3  |-  ( A. x  y  =  x  <->  A. x  x  =  y )
1512, 14xchbinx 311 . 2  |-  ( E. x  -.  y  =  x  <->  -.  A. x  x  =  y )
1611, 15mpbi 211 1  |-  -.  A. x  x  =  y
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   A.wal 1435    = wceq 1437   E.wex 1659   (/)c0 3758   {csn 3993
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 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-v 3080  df-dif 3436  df-un 3438  df-nul 3759  df-sn 3994  df-pr 3996
This theorem is referenced by: (None)
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