MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  aevlem1 Structured version   Visualization version   Unicode version

Theorem aevlem1 2032
Description: Lemma for aev 2036 and axc16g 2033. Change free and bound variables. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-13 2101, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.)
Assertion
Ref Expression
aevlem1  |-  ( A. z  z  =  w  ->  A. y  y  =  x )
Distinct variable groups:    z, w    x, y

Proof of Theorem aevlem1
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 cbvaev 1896 . 2  |-  ( A. z  z  =  w  ->  A. v  v  =  w )
2 ax5d 1769 . . 3  |-  ( -. 
A. z  z  =  v  ->  ( v  =  w  ->  A. z 
v  =  w ) )
32axc11nlem 2031 . 2  |-  ( A. v  v  =  w  ->  A. z  z  =  v )
4 cbvaev 1896 . 2  |-  ( A. z  z  =  v  ->  A. x  x  =  v )
5 ax5d 1769 . . 3  |-  ( -. 
A. y  y  =  x  ->  ( x  =  v  ->  A. y  x  =  v )
)
65axc11nlem 2031 . 2  |-  ( A. x  x  =  v  ->  A. y  y  =  x )
71, 3, 4, 64syl 19 1  |-  ( A. z  z  =  w  ->  A. y  y  =  x )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-12 1943
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674
This theorem is referenced by:  axc16g  2033  aev  2036  aevALT  2165
  Copyright terms: Public domain W3C validator