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Theorem ax10from10o 2129
Description: Rederivation of ax-10 2092 from original version ax-10o 2091. See theorem ax10o 1905 for the derivation of ax-10o 2091 from ax-10 2092.

This theorem should not be referenced in any proof. Instead, use ax-10 2092 above so that uses of ax-10 2092 can be more easily identified, or use aecom-o 2103 when this form is needed for studies involving ax-10o 2091 and omitting ax-17 1606. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion
Ref Expression
ax10from10o  |-  ( A. x  x  =  y  ->  A. y  y  =  x )

Proof of Theorem ax10from10o
StepHypRef Expression
1 ax-10o 2091 . . 3  |-  ( A. x  x  =  y  ->  ( A. x  x  =  y  ->  A. y  x  =  y )
)
21pm2.43i 43 . 2  |-  ( A. x  x  =  y  ->  A. y  x  =  y )
3 equcomi 1664 . . 3  |-  ( x  =  y  ->  y  =  x )
43alimi 1549 . 2  |-  ( A. y  x  =  y  ->  A. y  y  =  x )
52, 4syl 15 1  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4   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-10o 2091
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