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Theorem ax10lem6 1883
Description: Lemma for ax10 1884. Similar to ax10o 1892 but with reversed antecedent. (Contributed by NM, 25-Jul-2015.)
Assertion
Ref Expression
ax10lem6  |-  ( A. y  y  =  x  ->  ( A. x ph  ->  A. y ph )
)

Proof of Theorem ax10lem6
StepHypRef Expression
1 ax-11 1715 . . 3  |-  ( y  =  x  ->  ( A. x ph  ->  A. y
( y  =  x  ->  ph ) ) )
21sps 1739 . 2  |-  ( A. y  y  =  x  ->  ( A. x ph  ->  A. y ( y  =  x  ->  ph )
) )
3 pm2.27 35 . . 3  |-  ( y  =  x  ->  (
( y  =  x  ->  ph )  ->  ph )
)
43al2imi 1548 . 2  |-  ( A. y  y  =  x  ->  ( A. y ( y  =  x  ->  ph )  ->  A. y ph ) )
52, 4syld 40 1  |-  ( A. y  y  =  x  ->  ( A. x ph  ->  A. y ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527
This theorem is referenced by:  ax10  1884  a12stdy1-x12  29111  a12stdy2-x12  29112
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-11 1715
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