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Theorem dtruALT 4360
Description: A version of dtru 4354 ("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 3007 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 4335 . . . 4  |-  ( y  =  (/)  ->  -.  y  =  { (/) } )
2 p0ex 4350 . . . . 5  |-  { (/) }  e.  _V
3 eqeq2 2417 . . . . . 6  |-  ( x  =  { (/) }  ->  ( y  =  x  <->  y  =  { (/) } ) )
43notbid 286 . . . . 5  |-  ( x  =  { (/) }  ->  ( -.  y  =  x  <->  -.  y  =  { (/)
} ) )
52, 4spcev 3007 . . . 4  |-  ( -.  y  =  { (/) }  ->  E. x  -.  y  =  x )
61, 5syl 16 . . 3  |-  ( y  =  (/)  ->  E. x  -.  y  =  x
)
7 0ex 4303 . . . 4  |-  (/)  e.  _V
8 eqeq2 2417 . . . . 5  |-  ( x  =  (/)  ->  ( y  =  x  <->  y  =  (/) ) )
98notbid 286 . . . 4  |-  ( x  =  (/)  ->  ( -.  y  =  x  <->  -.  y  =  (/) ) )
107, 9spcev 3007 . . 3  |-  ( -.  y  =  (/)  ->  E. x  -.  y  =  x
)
116, 10pm2.61i 158 . 2  |-  E. x  -.  y  =  x
12 exnal 1580 . . 3  |-  ( E. x  -.  y  =  x  <->  -.  A. x  y  =  x )
13 eqcom 2410 . . . 4  |-  ( y  =  x  <->  x  =  y )
1413albii 1572 . . 3  |-  ( A. x  y  =  x  <->  A. x  x  =  y )
1512, 14xchbinx 302 . 2  |-  ( E. x  -.  y  =  x  <->  -.  A. x  x  =  y )
1611, 15mpbi 200 1  |-  -.  A. x  x  =  y
Colors of variables: wff set class
Syntax hints:   -. wn 3   A.wal 1546   E.wex 1547    = wceq 1649   (/)c0 3592   {csn 3778
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-v 2922  df-dif 3287  df-in 3291  df-ss 3298  df-nul 3593  df-pw 3765  df-sn 3784
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