MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax12olem4 Unicode version

Theorem ax12olem4 1883
Description: Lemma for ax12o 1887. Construct an intermediate equivalent to ax-12 1878 from two instances of ax-12 1878. (Contributed by NM, 24-Dec-2015.)
Hypotheses
Ref Expression
ax12olem4.1  |-  ( -.  x  =  y  -> 
( y  =  z  ->  A. x  y  =  z ) )
ax12olem4.2  |-  ( -.  x  =  y  -> 
( y  =  w  ->  A. x  y  =  w ) )
Assertion
Ref Expression
ax12olem4  |-  ( -.  x  =  y  -> 
( -.  A. x  -.  y  =  z  ->  A. x  y  =  z ) )
Distinct variable groups:    x, w, z    y, w, z

Proof of Theorem ax12olem4
StepHypRef Expression
1 ax12olem4.1 . 2  |-  ( -.  x  =  y  -> 
( y  =  z  ->  A. x  y  =  z ) )
2 ax12olem4.2 . . 3  |-  ( -.  x  =  y  -> 
( y  =  w  ->  A. x  y  =  w ) )
32ax12olem2 1881 . 2  |-  ( -.  x  =  y  -> 
( -.  y  =  z  ->  A. x  -.  y  =  z
) )
4 ax12olem3 1882 . 2  |-  ( ( -.  x  =  y  ->  ( -.  A. x  -.  y  =  z  ->  A. x  y  =  z ) )  <->  ( ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z ) )  /\  ( -.  x  =  y  ->  ( -.  y  =  z  ->  A. x  -.  y  =  z
) ) ) )
51, 3, 4mpbir2an 886 1  |-  ( -.  x  =  y  -> 
( -.  A. x  -.  y  =  z  ->  A. x  y  =  z ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1530
This theorem is referenced by:  ax12o  1887
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
  Copyright terms: Public domain W3C validator