MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax10lem5 Unicode version

Theorem ax10lem5 1974
Description: Lemma for ax10 1976. Change free and bound variables. (Contributed by NM, 22-Jul-2015.)
Assertion
Ref Expression
ax10lem5  |-  ( A. z  z  =  w  ->  A. y  y  =  x )
Distinct variable group:    z, w

Proof of Theorem ax10lem5
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax10lem1 1969 . . . 4  |-  ( A. z  z  =  w  ->  A. v  v  =  w )
2 ax10lem4 1973 . . . 4  |-  ( A. v  v  =  w  ->  A. u  u  =  v )
31, 2syl 16 . . 3  |-  ( A. z  z  =  w  ->  A. u  u  =  v )
4 ax10lem1 1969 . . 3  |-  ( A. u  u  =  v  ->  A. x  x  =  v )
53, 4syl 16 . 2  |-  ( A. z  z  =  w  ->  A. x  x  =  v )
6 ax10lem4 1973 . 2  |-  ( A. x  x  =  v  ->  A. y  y  =  x )
75, 6syl 16 1  |-  ( A. z  z  =  w  ->  A. y  y  =  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1546
This theorem is referenced by:  ax10  1976  a16g  1977  aev  2017
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551
  Copyright terms: Public domain W3C validator