MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax12olem1 Unicode version

Theorem ax12olem1 1880
Description: Lemma for ax12o 1887. Similar to equvin 1954 but with a negated equality. (Contributed by NM, 24-Dec-2015.)
Assertion
Ref Expression
ax12olem1  |-  ( E. w ( y  =  w  /\  -.  z  =  w )  <->  -.  y  =  z )
Distinct variable groups:    y, w    z, w

Proof of Theorem ax12olem1
StepHypRef Expression
1 ax-8 1661 . . . . 5  |-  ( y  =  w  ->  (
y  =  z  ->  w  =  z )
)
2 equcomi 1664 . . . . 5  |-  ( w  =  z  ->  z  =  w )
31, 2syl6 29 . . . 4  |-  ( y  =  w  ->  (
y  =  z  -> 
z  =  w ) )
43con3and 428 . . 3  |-  ( ( y  =  w  /\  -.  z  =  w
)  ->  -.  y  =  z )
54exlimiv 1624 . 2  |-  ( E. w ( y  =  w  /\  -.  z  =  w )  ->  -.  y  =  z )
6 ax-17 1606 . . 3  |-  ( -.  y  =  z  ->  A. w  -.  y  =  z )
7 ax-8 1661 . . . . . . . 8  |-  ( w  =  z  ->  (
w  =  y  -> 
z  =  y ) )
8 equcomi 1664 . . . . . . . 8  |-  ( z  =  y  ->  y  =  z )
97, 8syl6 29 . . . . . . 7  |-  ( w  =  z  ->  (
w  =  y  -> 
y  =  z ) )
109equcoms 1666 . . . . . 6  |-  ( z  =  w  ->  (
w  =  y  -> 
y  =  z ) )
1110com12 27 . . . . 5  |-  ( w  =  y  ->  (
z  =  w  -> 
y  =  z ) )
1211con3d 125 . . . 4  |-  ( w  =  y  ->  ( -.  y  =  z  ->  -.  z  =  w ) )
13 equcomi 1664 . . . 4  |-  ( w  =  y  ->  y  =  w )
1412, 13jctild 527 . . 3  |-  ( w  =  y  ->  ( -.  y  =  z  ->  ( y  =  w  /\  -.  z  =  w ) ) )
156, 14spimeh 1734 . 2  |-  ( -.  y  =  z  ->  E. w ( y  =  w  /\  -.  z  =  w ) )
165, 15impbii 180 1  |-  ( E. w ( y  =  w  /\  -.  z  =  w )  <->  -.  y  =  z )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531
This theorem is referenced by:  ax12olem2  1881  ax12olem2wAUX7  29432  ax12olem2OLD7  29660
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-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
  Copyright terms: Public domain W3C validator