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Theorem ax10lem3 1878
Description: Lemma for ax10 1884. Similar to ax-10 2079 but with distinct variables. (Contributed by NM, 25-Jul-2015.)
Assertion
Ref Expression
ax10lem3  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
Distinct variable group:    x, y

Proof of Theorem ax10lem3
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax10lem2 1877 . 2  |-  ( A. x  x  =  y  ->  A. x  x  =  z )
2 ax10lem1 1876 . . . 4  |-  ( A. x  x  =  z  ->  A. w  w  =  z )
3 ax10lem2 1877 . . . 4  |-  ( A. w  w  =  z  ->  A. w  w  =  x )
42, 3syl 15 . . 3  |-  ( A. x  x  =  z  ->  A. w  w  =  x )
5 ax10lem1 1876 . . 3  |-  ( A. w  w  =  x  ->  A. y  y  =  x )
64, 5syl 15 . 2  |-  ( A. x  x  =  z  ->  A. y  y  =  x )
71, 6syl 15 1  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527
This theorem is referenced by:  dvelimv  1879  hbae-x12  28482  a12stdy1-x12  28484  a12stdy2-x12  28485  ax10lem17ALT  28496
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529
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