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Theorem dtruALT 4399
Description: A version of dtru 4393 ("two things exist") with a shorter proof that uses more axioms but may be easier to understand.

Assuming that ZF set theory is consistent, we cannot prove this theorem unless we specify that  x and  y be distinct. Specifically, theorem spcev 3045 requires that  x must not occur in the subexpression  -.  y  =  { (/) } in step 4 nor in the subexpression  -.  y  =  (/) in step 9. The proof verifier will require that  x and  y be in a distinct variable group to ensure this. You can check this by deleting the $d statement in set.mm and rerunning the verifier, which will print a detailed explanation of the distinct variable violation. (Contributed by NM, 15-Jul-1994.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion
Ref Expression
dtruALT  |-  -.  A. x  x  =  y
Distinct variable group:    x, y

Proof of Theorem dtruALT
StepHypRef Expression
1 0inp0 4374 . . . 4  |-  ( y  =  (/)  ->  -.  y  =  { (/) } )
2 p0ex 4389 . . . . 5  |-  { (/) }  e.  _V
3 eqeq2 2447 . . . . . 6  |-  ( x  =  { (/) }  ->  ( y  =  x  <->  y  =  { (/) } ) )
43notbid 287 . . . . 5  |-  ( x  =  { (/) }  ->  ( -.  y  =  x  <->  -.  y  =  { (/)
} ) )
52, 4spcev 3045 . . . 4  |-  ( -.  y  =  { (/) }  ->  E. x  -.  y  =  x )
61, 5syl 16 . . 3  |-  ( y  =  (/)  ->  E. x  -.  y  =  x
)
7 0ex 4342 . . . 4  |-  (/)  e.  _V
8 eqeq2 2447 . . . . 5  |-  ( x  =  (/)  ->  ( y  =  x  <->  y  =  (/) ) )
98notbid 287 . . . 4  |-  ( x  =  (/)  ->  ( -.  y  =  x  <->  -.  y  =  (/) ) )
107, 9spcev 3045 . . 3  |-  ( -.  y  =  (/)  ->  E. x  -.  y  =  x
)
116, 10pm2.61i 159 . 2  |-  E. x  -.  y  =  x
12 exnal 1584 . . 3  |-  ( E. x  -.  y  =  x  <->  -.  A. x  y  =  x )
13 eqcom 2440 . . . 4  |-  ( y  =  x  <->  x  =  y )
1413albii 1576 . . 3  |-  ( A. x  y  =  x  <->  A. x  x  =  y )
1512, 14xchbinx 303 . 2  |-  ( E. x  -.  y  =  x  <->  -.  A. x  x  =  y )
1611, 15mpbi 201 1  |-  -.  A. x  x  =  y
Colors of variables: wff set class
Syntax hints:   -. wn 3   A.wal 1550   E.wex 1551    = wceq 1653   (/)c0 3630   {csn 3816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2960  df-dif 3325  df-in 3329  df-ss 3336  df-nul 3631  df-pw 3803  df-sn 3822
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