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Theorem sb9 2265
Description: Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) Allow a shortening of sb9i 2266. (Revised by Wolf Lammen, 15-Jun-2019.)
Assertion
Ref Expression
sb9  |-  ( A. x [ x  /  y ] ph  <->  A. y [ y  /  x ] ph )

Proof of Theorem sb9
StepHypRef Expression
1 sbequ12a 2095 . . . . 5  |-  ( y  =  x  ->  ( [ x  /  y ] ph  <->  [ y  /  x ] ph ) )
21equcoms 1874 . . . 4  |-  ( x  =  y  ->  ( [ x  /  y ] ph  <->  [ y  /  x ] ph ) )
32sps 1953 . . 3  |-  ( A. x  x  =  y  ->  ( [ x  / 
y ] ph  <->  [ y  /  x ] ph )
)
43dral1 2169 . 2  |-  ( A. x  x  =  y  ->  ( A. x [
x  /  y ]
ph 
<-> 
A. y [ y  /  x ] ph ) )
5 nfnae 2162 . . 3  |-  F/ x  -.  A. x  x  =  y
6 nfnae 2162 . . 3  |-  F/ y  -.  A. x  x  =  y
7 nfsb2 2200 . . . 4  |-  ( -. 
A. y  y  =  x  ->  F/ y [ x  /  y ] ph )
87naecoms 2157 . . 3  |-  ( -. 
A. x  x  =  y  ->  F/ y [ x  /  y ] ph )
9 nfsb2 2200 . . 3  |-  ( -. 
A. x  x  =  y  ->  F/ x [ y  /  x ] ph )
102a1i 11 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( [ x  /  y ]
ph 
<->  [ y  /  x ] ph ) ) )
115, 6, 8, 9, 10cbv2 2123 . 2  |-  ( -. 
A. x  x  =  y  ->  ( A. x [ x  /  y ] ph  <->  A. y [ y  /  x ] ph ) )
124, 11pm2.61i 169 1  |-  ( A. x [ x  /  y ] ph  <->  A. y [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189   A.wal 1452   F/wnf 1677   [wsb 1807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-nf 1678  df-sb 1808
This theorem is referenced by:  sb9i  2266
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