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

Theorem axc11 2110
Description: Show that ax-c11 32168 can be derived from ax-c11n 32169 in the form of axc11n 2105. Normally, axc11 2110 should be used rather than ax-c11 32168, 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 1995 . 2  |-  ( A. y  y  =  x  ->  ( A. x ph  ->  A. y ph )
)
21aecoms 2108 1  |-  ( A. x  x  =  y  ->  ( 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 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-12 1907  ax-13 2055
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664
This theorem is referenced by:  hbae  2111  axc16gOLD  2117  dral1  2123  dral1ALT  2124  nd1  9010  nd2  9011  bj-hbaeb2  31171  wl-aetr  31570  ax6e2eq  36561  ax6e2eqVD  36944  2sb5ndVD  36947  2sb5ndALT  36969
  Copyright terms: Public domain W3C validator