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Theorem dtruALT 4632
Description: Alternate proof of dtru 4594 which requires more axioms but is shorter and 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 3141 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 4575 . . . 4  |-  ( y  =  (/)  ->  -.  y  =  { (/) } )
2 p0ex 4590 . . . . 5  |-  { (/) }  e.  _V
3 eqeq2 2462 . . . . . 6  |-  ( x  =  { (/) }  ->  ( y  =  x  <->  y  =  { (/) } ) )
43notbid 296 . . . . 5  |-  ( x  =  { (/) }  ->  ( -.  y  =  x  <->  -.  y  =  { (/)
} ) )
52, 4spcev 3141 . . . 4  |-  ( -.  y  =  { (/) }  ->  E. x  -.  y  =  x )
61, 5syl 17 . . 3  |-  ( y  =  (/)  ->  E. x  -.  y  =  x
)
7 0ex 4535 . . . 4  |-  (/)  e.  _V
8 eqeq2 2462 . . . . 5  |-  ( x  =  (/)  ->  ( y  =  x  <->  y  =  (/) ) )
98notbid 296 . . . 4  |-  ( x  =  (/)  ->  ( -.  y  =  x  <->  -.  y  =  (/) ) )
107, 9spcev 3141 . . 3  |-  ( -.  y  =  (/)  ->  E. x  -.  y  =  x
)
116, 10pm2.61i 168 . 2  |-  E. x  -.  y  =  x
12 exnal 1699 . . 3  |-  ( E. x  -.  y  =  x  <->  -.  A. x  y  =  x )
13 eqcom 2458 . . . 4  |-  ( y  =  x  <->  x  =  y )
1413albii 1691 . . 3  |-  ( A. x  y  =  x  <->  A. x  x  =  y )
1512, 14xchbinx 312 . 2  |-  ( E. x  -.  y  =  x  <->  -.  A. x  x  =  y )
1611, 15mpbi 212 1  |-  -.  A. x  x  =  y
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   A.wal 1442    = wceq 1444   E.wex 1663   (/)c0 3731   {csn 3968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-v 3047  df-dif 3407  df-in 3411  df-ss 3418  df-nul 3732  df-pw 3953  df-sn 3969
This theorem is referenced by: (None)
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