MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axc11 Structured version   Unicode version

Theorem axc11 2027
Description: Show that ax-c11 2211 can be derived from ax-c11n 2212 in the form of axc11n 2022. Normally, axc11 2027 should be used rather than ax-c11 2211, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof shortened by Wolf Lammen, 21-Apr-2018.)
Assertion
Ref Expression
axc11  |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ph )
)

Proof of Theorem axc11
StepHypRef Expression
1 axc112 1884 . 2  |-  ( A. y  y  =  x  ->  ( A. x ph  ->  A. y ph )
)
21aecoms 2025 1  |-  ( A. x  x  =  y  ->  ( 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-10 1786  ax-12 1803  ax-13 1968
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600
This theorem is referenced by:  hbae  2028  axc16gOLD  2034  dral1  2040  dral1ALT  2041  nd1  8961  nd2  8962  axpowndlem3OLD  8975  wl-aetr  29576  ax6e2eq  32419  ax6e2eqVD  32796  2sb5ndVD  32799  2sb5ndALT  32821  bj-hbaeb2  33481
  Copyright terms: Public domain W3C validator