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Theorem ax9OLD 29781
Description: Theorem showing that ax9 1902 follows from the weaker version ax9v 1645.

See also ax9 1902 for a slightly more direct proof (using lemmas for ax10 1897 derivation).

This theorem normally should not be referenced in any later proof. Instead, the use of ax9 1902 below is preferred, since it is easier to work with (it has no distinct variable conditions) and it is the standard version we have adopted. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion
Ref Expression
ax9OLD  |-  -.  A. x  -.  x  =  y

Proof of Theorem ax9OLD
Dummy variables  u  t  v  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax9v 1645 . 2  |-  -.  A. t  -.  t  =  u
2 ax9v 1645 . 2  |-  -.  A. t  -.  t  =  z
3 ax9v 1645 . 2  |-  -.  A. u  -.  u  =  x
4 ax9v 1645 . 2  |-  -.  A. u  -.  u  =  w
5 ax9v 1645 . 2  |-  -.  A. z  -.  z  =  x
6 ax9v 1645 . 2  |-  -.  A. z  -.  z  =  w
7 ax9v 1645 . 2  |-  -.  A. x  -.  x  =  t
8 ax9v 1645 . 2  |-  -.  A. x  -.  x  =  z
9 ax9v 1645 . 2  |-  -.  A. w  -.  w  =  t
10 ax9v 1645 . 2  |-  -.  A. w  -.  w  =  z
11 ax9v 1645 . 2  |-  -.  A. w  -.  w  =  y
12 ax9v 1645 . 2  |-  -.  A. x  -.  x  =  v
13 ax9v 1645 . 2  |-  -.  A. v  -.  v  =  y
141, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13ax9vax9 29780 1  |-  -.  A. x  -.  x  =  y
Colors of variables: wff set class
Syntax hints:   -. wn 3   A.wal 1530
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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