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Theorem dtruALT2 4235
Description: An alternative proof of dtru 4217 ("two things exist") using ax-pr 4230 instead of ax-pow 4204. (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 4198 . . . 4  |-  ( y  =  (/)  ->  -.  y  =  { (/) } )
2 snex 4232 . . . . 5  |-  { (/) }  e.  _V
3 eqeq2 2305 . . . . . 6  |-  ( x  =  { (/) }  ->  ( y  =  x  <->  y  =  { (/) } ) )
43notbid 285 . . . . 5  |-  ( x  =  { (/) }  ->  ( -.  y  =  x  <->  -.  y  =  { (/)
} ) )
52, 4spcev 2888 . . . 4  |-  ( -.  y  =  { (/) }  ->  E. x  -.  y  =  x )
61, 5syl 15 . . 3  |-  ( y  =  (/)  ->  E. x  -.  y  =  x
)
7 0ex 4166 . . . 4  |-  (/)  e.  _V
8 eqeq2 2305 . . . . 5  |-  ( x  =  (/)  ->  ( y  =  x  <->  y  =  (/) ) )
98notbid 285 . . . 4  |-  ( x  =  (/)  ->  ( -.  y  =  x  <->  -.  y  =  (/) ) )
107, 9spcev 2888 . . 3  |-  ( -.  y  =  (/)  ->  E. x  -.  y  =  x
)
116, 10pm2.61i 156 . 2  |-  E. x  -.  y  =  x
12 exnal 1564 . . 3  |-  ( E. x  -.  y  =  x  <->  -.  A. x  y  =  x )
13 eqcom 2298 . . . 4  |-  ( y  =  x  <->  x  =  y )
1413albii 1556 . . 3  |-  ( A. x  y  =  x  <->  A. x  x  =  y )
1512, 14xchbinx 301 . 2  |-  ( E. x  -.  y  =  x  <->  -.  A. x  x  =  y )
1611, 15mpbi 199 1  |-  -.  A. x  x  =  y
Colors of variables: wff set class
Syntax hints:   -. wn 3   A.wal 1530   E.wex 1531    = wceq 1632   (/)c0 3468   {csn 3653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-nul 3469  df-sn 3659  df-pr 3660
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