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Theorem axc112 2019
Description: Same as axc11 2147 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 1932 . . 3  |-  ( y  =  x  ->  ( A. x ph  ->  A. y
( y  =  x  ->  ph ) ) )
21sps 1942 . 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 1686 . 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 1441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-12 1932
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1663
This theorem is referenced by:  axc16g  2022  axc11n  2142  axc11nALT  2143  axc11  2147  hbae  2148  dral1  2158  dral1ALT  2159  axpowndlem3  9021  bj-axc11nv  31341  bj-axc11v  31344  bj-dral1v  31351  bj-hbaeb2  31413
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