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Theorem ax12olem1 2005
Description: Lemma for nfeqf 2009 and dveeq1 2021. Used to eliminate distinct variable constraints. The proof of ax12o 2010 bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 8-Feb-2018.)
Assertion
Ref Expression
ax12olem1  |-  ( y  =  z  <->  A. w
( y  =  w  ->  z  =  w ) )
Distinct variable groups:    y, w    z, w

Proof of Theorem ax12olem1
StepHypRef Expression
1 ax-8 1687 . . 3  |-  ( y  =  z  ->  (
y  =  w  -> 
z  =  w ) )
21alrimiv 1641 . 2  |-  ( y  =  z  ->  A. w
( y  =  w  ->  z  =  w ) )
3 equcomi 1691 . . . 4  |-  ( w  =  y  ->  y  =  w )
4 equcomi 1691 . . . . 5  |-  ( z  =  w  ->  w  =  z )
5 ax-8 1687 . . . . 5  |-  ( w  =  y  ->  (
w  =  z  -> 
y  =  z ) )
64, 5syl5 30 . . . 4  |-  ( w  =  y  ->  (
z  =  w  -> 
y  =  z ) )
73, 6embantd 52 . . 3  |-  ( w  =  y  ->  (
( y  =  w  ->  z  =  w )  ->  y  =  z ) )
87spimvw 1681 . 2  |-  ( A. w ( y  =  w  ->  z  =  w )  ->  y  =  z )
92, 8impbii 181 1  |-  ( y  =  z  <->  A. w
( y  =  w  ->  z  =  w ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549
This theorem is referenced by:  ax12olem2  2006  ax12olem4  2008
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551
  Copyright terms: Public domain W3C validator