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Theorem axc112 1992
Description: Same as axc11 2107 but with reversed antecedent. (Contributed by NM, 25-Jul-2015.)
Assertion
Ref Expression
axc112  |-  ( A. y  y  =  x  ->  ( A. x ph  ->  A. y ph )
)

Proof of Theorem axc112
StepHypRef Expression
1 ax-12 1904 . . 3  |-  ( y  =  x  ->  ( A. x ph  ->  A. y
( y  =  x  ->  ph ) ) )
21sps 1915 . 2  |-  ( A. y  y  =  x  ->  ( A. x ph  ->  A. y ( y  =  x  ->  ph )
) )
3 pm2.27 40 . . 3  |-  ( y  =  x  ->  (
( y  =  x  ->  ph )  ->  ph )
)
43al2imi 1683 . 2  |-  ( A. y  y  =  x  ->  ( A. y ( y  =  x  ->  ph )  ->  A. y ph ) )
52, 4syld 45 1  |-  ( A. y  y  =  x  ->  ( A. x ph  ->  A. y ph )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-12 1904
This theorem depends on definitions:  df-bi 188  df-ex 1660
This theorem is referenced by:  axc16g  1995  axc11n  2102  axc11nOLD  2103  axc11  2107  hbae  2108  dral1  2120  dral1ALT  2121  axpowndlem3  9013  bj-axc11nv  31137  bj-axc11v  31140  bj-dral1v  31147  bj-hbaeb2  31212
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