MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax10lem16 Unicode version

Theorem ax10lem16 1665
Description: Lemma for ax10 1677. Similar to equequ2 1830, without using ax-4 1692, ax-9 1684, ax-10 1678, or ax-12o 1664 but allowing ax-9v 1632. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.)
Assertion
Ref Expression
ax10lem16  |-  ( x  =  y  ->  (
z  =  x  <->  z  =  y ) )

Proof of Theorem ax10lem16
StepHypRef Expression
1 ax12o10lem1 1635 . . 3  |-  ( z  =  x  ->  x  =  z )
2 ax-8 1623 . . . 4  |-  ( x  =  z  ->  (
x  =  y  -> 
z  =  y ) )
32com12 29 . . 3  |-  ( x  =  y  ->  (
x  =  z  -> 
z  =  y ) )
41, 3syl5 30 . 2  |-  ( x  =  y  ->  (
z  =  x  -> 
z  =  y ) )
5 ax12o10lem1 1635 . . 3  |-  ( x  =  y  ->  y  =  x )
6 ax12o10lem1 1635 . . 3  |-  ( z  =  y  ->  y  =  z )
7 ax-8 1623 . . . 4  |-  ( y  =  z  ->  (
y  =  x  -> 
z  =  x ) )
87com12 29 . . 3  |-  ( y  =  x  ->  (
y  =  z  -> 
z  =  x ) )
95, 6, 8syl2im 36 . 2  |-  ( x  =  y  ->  (
z  =  y  -> 
z  =  x ) )
104, 9impbid 185 1  |-  ( x  =  y  ->  (
z  =  x  <->  z  =  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178
This theorem is referenced by:  ax10lem21  1670  ax10lem25  1674  ax10lem26  1675  ax9  1683  ax10lem21X  27891  ax10lem25X  27895  ax10lem26X  27896  ax9X  27903  a12stdy2-x12  27913  a12study4  27918  ax10lem17ALT  27924  ax10lem18ALT  27925
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-gen 1536  ax-8 1623  ax-17 1628  ax-9v 1632
This theorem depends on definitions:  df-bi 179
  Copyright terms: Public domain W3C validator