MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axc112 Structured version   Unicode version

Theorem axc112 1884
Description: Same as axc11 2027 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 1803 . . 3  |-  ( y  =  x  ->  ( A. x ph  ->  A. y
( y  =  x  ->  ph ) ) )
21sps 1814 . 2  |-  ( A. y  y  =  x  ->  ( A. x ph  ->  A. y ( y  =  x  ->  ph )
) )
3 pm2.27 39 . . 3  |-  ( y  =  x  ->  (
( y  =  x  ->  ph )  ->  ph )
)
43al2imi 1616 . 2  |-  ( A. y  y  =  x  ->  ( A. y ( y  =  x  ->  ph )  ->  A. y ph ) )
52, 4syld 44 1  |-  ( A. y  y  =  x  ->  ( A. x ph  ->  A. y ph )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-12 1803
This theorem depends on definitions:  df-bi 185  df-ex 1597
This theorem is referenced by:  axc16g  1887  axc11n  2022  axc11nOLD  2023  axc11  2027  hbae  2028  dral1  2040  dral1ALT  2041  axpowndlem3  8976  bj-axc11nv  33614  bj-axc11v  33617  bj-dral1v  33624  bj-hbaeb2  33689
  Copyright terms: Public domain W3C validator