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Theorem ax10from10o 2255
Description: Rederivation of ax-10 2218 from original version ax-10o 2217. See theorem ax10o 2039 for the derivation of ax-10o 2217 from ax-10 2218.

This theorem should not be referenced in any proof. Instead, use ax-10 2218 above so that uses of ax-10 2218 can be more easily identified, or use aecom-o 2229 when this form is needed for studies involving ax-10o 2217 and omitting ax-17 1627. (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 2217 . . 3  |-  ( A. x  x  =  y  ->  ( A. x  x  =  y  ->  A. y  x  =  y )
)
21pm2.43i 46 . 2  |-  ( A. x  x  =  y  ->  A. y  x  =  y )
3 equcomi 1692 . . 3  |-  ( x  =  y  ->  y  =  x )
43alimi 1569 . 2  |-  ( A. y  x  =  y  ->  A. y  y  =  x )
52, 4syl 16 1  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-10o 2217
This theorem depends on definitions:  df-bi 179  df-ex 1552
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