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Theorem ax10lem2 2023
Description: Lemma for ax10 2025. Change bound variable. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.)
Assertion
Ref Expression
ax10lem2  |-  ( A. x  x  =  w  ->  A. y  y  =  x )
Distinct variable groups:    x, w    y, w

Proof of Theorem ax10lem2
StepHypRef Expression
1 ax10lem1 2022 . . 3  |-  ( A. x  x  =  w  ->  A. y  y  =  w )
2 equequ2 1698 . . . . 5  |-  ( x  =  w  ->  (
y  =  x  <->  y  =  w ) )
32biimprd 215 . . . 4  |-  ( x  =  w  ->  (
y  =  w  -> 
y  =  x ) )
43al2imi 1570 . . 3  |-  ( A. y  x  =  w  ->  ( A. y  y  =  w  ->  A. y 
y  =  x ) )
51, 4syl5com 28 . 2  |-  ( A. x  x  =  w  ->  ( A. y  x  =  w  ->  A. y 
y  =  x ) )
6 dveeq1 2021 . . . . 5  |-  ( -. 
A. y  y  =  x  ->  ( x  =  w  ->  A. y  x  =  w )
)
76spsd 1771 . . . 4  |-  ( -. 
A. y  y  =  x  ->  ( A. x  x  =  w  ->  A. y  x  =  w ) )
87com12 29 . . 3  |-  ( A. x  x  =  w  ->  ( -.  A. y 
y  =  x  ->  A. y  x  =  w ) )
98con1d 118 . 2  |-  ( A. x  x  =  w  ->  ( -.  A. y  x  =  w  ->  A. y  y  =  x ) )
105, 9pm2.61d 152 1  |-  ( A. x  x  =  w  ->  A. y  y  =  x )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1549
This theorem is referenced by:  ax10  2025  aevlem1  2046
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  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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