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Theorem axc11nlem 2021
Description: Lemma for axc11n 2143. Change bound variable in an equality. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Restructure to ease either bundling, or reducing dependencies on axioms. (Revised by Wolf Lammen, 30-Nov-2019.)
Hypothesis
Ref Expression
axc11nlem.1  |-  ( -. 
A. y  y  =  x  ->  ( x  =  z  ->  A. y  x  =  z )
)
Assertion
Ref Expression
axc11nlem  |-  ( A. x  x  =  z  ->  A. y  y  =  x )
Distinct variable groups:    x, z    y, z

Proof of Theorem axc11nlem
StepHypRef Expression
1 cbvaev 1886 . . 3  |-  ( A. x  x  =  z  ->  A. y  y  =  z )
2 equequ2 1868 . . . . 5  |-  ( x  =  z  ->  (
y  =  x  <->  y  =  z ) )
32biimprd 227 . . . 4  |-  ( x  =  z  ->  (
y  =  z  -> 
y  =  x ) )
43al2imi 1687 . . 3  |-  ( A. y  x  =  z  ->  ( A. y  y  =  z  ->  A. y 
y  =  x ) )
51, 4syl5com 31 . 2  |-  ( A. x  x  =  z  ->  ( A. y  x  =  z  ->  A. y 
y  =  x ) )
6 axc11nlem.1 . . . . 5  |-  ( -. 
A. y  y  =  x  ->  ( x  =  z  ->  A. y  x  =  z )
)
76spsd 1945 . . . 4  |-  ( -. 
A. y  y  =  x  ->  ( A. x  x  =  z  ->  A. y  x  =  z ) )
87com12 32 . . 3  |-  ( A. x  x  =  z  ->  ( -.  A. y 
y  =  x  ->  A. y  x  =  z ) )
98con1d 128 . 2  |-  ( A. x  x  =  z  ->  ( -.  A. y  x  =  z  ->  A. y  y  =  x ) )
105, 9pm2.61d 162 1  |-  ( A. x  x  =  z  ->  A. y  y  =  x )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-12 1933
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664
This theorem is referenced by:  aevlem1  2022  axc11n  2143
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