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Theorem axnulALT 4524
Description: Alternate proof of axnul 4525, proved directly from ax-rep 4508 using none of the equality axioms ax-7 1859 through ax-c14 32527 provided we accept sp 1957 as an axiom. Replace sp 1957 with the obsolete ax-c5 32519 to see this in 'show traceback'. (Contributed by Jeff Hoffman, 3-Feb-2008.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
axnulALT  |-  E. x A. y  -.  y  e.  x
Distinct variable group:    x, y

Proof of Theorem axnulALT
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-rep 4508 . . 3  |-  ( A. w E. x A. y
( A. x F.  ->  y  =  x )  ->  E. x A. y
( y  e.  x  <->  E. w ( w  e.  z  /\  A. x F.  ) ) )
2 sp 1957 . . . . . 6  |-  ( A. x  -.  A. y ( A. x F.  ->  y  =  x )  ->  -.  A. y ( A. x F.  ->  y  =  x ) )
32con2i 124 . . . . 5  |-  ( A. y ( A. x F.  ->  y  =  x )  ->  -.  A. x  -.  A. y ( A. x F.  ->  y  =  x ) )
4 df-ex 1672 . . . . 5  |-  ( E. x A. y ( A. x F.  ->  y  =  x )  <->  -.  A. x  -.  A. y ( A. x F.  ->  y  =  x ) )
53, 4sylibr 217 . . . 4  |-  ( A. y ( A. x F.  ->  y  =  x )  ->  E. x A. y ( A. x F.  ->  y  =  x ) )
6 fal 1459 . . . . . 6  |-  -. F.
7 sp 1957 . . . . . 6  |-  ( A. x F.  -> F.  )
86, 7mto 181 . . . . 5  |-  -.  A. x F.
98pm2.21i 136 . . . 4  |-  ( A. x F.  ->  y  =  x )
105, 9mpg 1679 . . 3  |-  E. x A. y ( A. x F.  ->  y  =  x )
111, 10mpg 1679 . 2  |-  E. x A. y ( y  e.  x  <->  E. w ( w  e.  z  /\  A. x F.  ) )
128intnan 928 . . . . . 6  |-  -.  (
w  e.  z  /\  A. x F.  )
1312nex 1686 . . . . 5  |-  -.  E. w ( w  e.  z  /\  A. x F.  )
1413nbn 354 . . . 4  |-  ( -.  y  e.  x  <->  ( y  e.  x  <->  E. w ( w  e.  z  /\  A. x F.  ) )
)
1514albii 1699 . . 3  |-  ( A. y  -.  y  e.  x  <->  A. y ( y  e.  x  <->  E. w ( w  e.  z  /\  A. x F.  ) )
)
1615exbii 1726 . 2  |-  ( E. x A. y  -.  y  e.  x  <->  E. x A. y ( y  e.  x  <->  E. w ( w  e.  z  /\  A. x F.  ) )
)
1711, 16mpbir 214 1  |-  E. x A. y  -.  y  e.  x
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376   A.wal 1450    = wceq 1452   F. wfal 1457   E.wex 1671    e. wcel 1904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-12 1950  ax-rep 4508
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-fal 1458  df-ex 1672
This theorem is referenced by: (None)
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