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Theorem axc11 2159
Description: Show that ax-c11 32504 can be derived from ax-c11n 32505 in the form of axc11n 2154. Normally, axc11 2159 should be used rather than ax-c11 32504, 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 2031 . 2  |-  ( A. y  y  =  x  ->  ( A. x ph  ->  A. y ph )
)
21aecoms 2157 1  |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ph )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-12 1944  ax-13 2102
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675  df-nf 1679
This theorem is referenced by:  hbae  2160  dral1  2170  dral1ALT  2171  nd1  9038  nd2  9039  bj-hbaeb2  31465  wl-aetr  31908  ax6e2eq  36968  ax6e2eqVD  37344  2sb5ndVD  37347  2sb5ndALT  37369
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