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Theorem ax9lem1 29140
Description: Lemma for ax9 1889. Similar to equcomi 1646, without using sp 1716, ax9 1889, or ax10 1884. (Contributed by NM, 7-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
ax9lem1.a  |-  -.  A. w  -.  w  =  x
Assertion
Ref Expression
ax9lem1  |-  ( x  =  y  ->  y  =  x )
Distinct variable group:    x, w

Proof of Theorem ax9lem1
StepHypRef Expression
1 ax9lem1.a . . . 4  |-  -.  A. w  -.  w  =  x
2 ax-8 1643 . . . . . . 7  |-  ( w  =  x  ->  (
w  =  x  ->  x  =  x )
)
32pm2.43i 43 . . . . . 6  |-  ( w  =  x  ->  x  =  x )
43con3i 127 . . . . 5  |-  ( -.  x  =  x  ->  -.  w  =  x
)
54alimi 1546 . . . 4  |-  ( A. w  -.  x  =  x  ->  A. w  -.  w  =  x )
61, 5mto 167 . . 3  |-  -.  A. w  -.  x  =  x
7 ax-17 1603 . . 3  |-  ( -.  x  =  x  ->  A. w  -.  x  =  x )
86, 7mt3 171 . 2  |-  x  =  x
9 ax-8 1643 . 2  |-  ( x  =  y  ->  (
x  =  x  -> 
y  =  x ) )
108, 9mpi 16 1  |-  ( x  =  y  ->  y  =  x )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1527
This theorem is referenced by:  ax9lem2  29141  ax9lem3  29142  ax9lem6  29145  ax9lem15  29154  ax9vax9  29158
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-8 1643
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