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Theorem ax9sep 29160
Description: Show that the Separation Axiom ax-sep 4141 and Extensionality ax-ext 2264 implies ax9 1889. Note that ax9 1889 and sp 1716 (which can be derived from ax9 1889) are not used by the proof. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax9sep  |-  -.  A. x  -.  x  =  y

Proof of Theorem ax9sep
Dummy variables  z  w  v  u  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax9vsep 4145 . 2  |-  -.  A. t  -.  t  =  u
2 ax9vsep 4145 . 2  |-  -.  A. t  -.  t  =  z
3 ax9vsep 4145 . 2  |-  -.  A. u  -.  u  =  x
4 ax9vsep 4145 . 2  |-  -.  A. u  -.  u  =  w
5 ax9vsep 4145 . 2  |-  -.  A. z  -.  z  =  x
6 ax9vsep 4145 . 2  |-  -.  A. z  -.  z  =  w
7 ax9vsep 4145 . 2  |-  -.  A. x  -.  x  =  t
8 ax9vsep 4145 . 2  |-  -.  A. x  -.  x  =  z
9 ax9vsep 4145 . 2  |-  -.  A. w  -.  w  =  t
10 ax9vsep 4145 . 2  |-  -.  A. w  -.  w  =  z
11 ax9vsep 4145 . 2  |-  -.  A. w  -.  w  =  y
12 ax9vsep 4145 . 2  |-  -.  A. x  -.  x  =  v
13 ax9vsep 4145 . 2  |-  -.  A. v  -.  v  =  y
141, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13ax9vax9 29158 1  |-  -.  A. x  -.  x  =  y
Colors of variables: wff set class
Syntax hints:   -. wn 3   A.wal 1527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529
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