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Theorem wl-equsb4 31849
Description: Substitution applied to an atomic wff. The distinctor antecedent is more general than a distinct variable constraint. (Contributed by Wolf Lammen, 26-Jun-2019.)
Assertion
Ref Expression
wl-equsb4  |-  ( -. 
A. x  x  =  z  ->  ( [
y  /  x ]
y  =  z  <->  y  =  z ) )

Proof of Theorem wl-equsb4
StepHypRef Expression
1 nfeqf 2104 . . . 4  |-  ( ( -.  A. x  x  =  y  /\  -.  A. x  x  =  z )  ->  F/ x  y  =  z )
21ex 435 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x  x  =  z  ->  F/ x  y  =  z ) )
3 sbft 2177 . . 3  |-  ( F/ x  y  =  z  ->  ( [ y  /  x ] y  =  z  <->  y  =  z ) )
42, 3syl6com 36 . 2  |-  ( -. 
A. x  x  =  z  ->  ( -.  A. x  x  =  y  ->  ( [ y  /  x ] y  =  z  <->  y  =  z ) ) )
5 sbequ12r 2052 . . . 4  |-  ( y  =  x  ->  ( [ y  /  x ] y  =  z  <-> 
y  =  z ) )
65equcoms 1849 . . 3  |-  ( x  =  y  ->  ( [ y  /  x ] y  =  z  <-> 
y  =  z ) )
76sps 1920 . 2  |-  ( A. x  x  =  y  ->  ( [ y  /  x ] y  =  z  <-> 
y  =  z ) )
84, 7pm2.61d2 163 1  |-  ( -. 
A. x  x  =  z  ->  ( [
y  /  x ]
y  =  z  <->  y  =  z ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187   A.wal 1435   F/wnf 1661   [wsb 1790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909  ax-13 2057
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662  df-sb 1791
This theorem is referenced by: (None)
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