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Theorem wl-sb8t 31844
Description: Substitution of variable in universal quantifier. Closed form of sb8 2222. (Contributed by Wolf Lammen, 27-Jul-2019.)
Assertion
Ref Expression
wl-sb8t  |-  ( A. x F/ y ph  ->  ( A. x ph  <->  A. y [ y  /  x ] ph ) )

Proof of Theorem wl-sb8t
StepHypRef Expression
1 nfa1 1956 . 2  |-  F/ x A. x F/ y ph
2 nfnf1 1958 . . 3  |-  F/ y F/ y ph
32nfal 2007 . 2  |-  F/ y A. x F/ y
ph
4 sp 1914 . 2  |-  ( A. x F/ y ph  ->  F/ y ph )
5 wl-nfs1t 31835 . . 3  |-  ( F/ y ph  ->  F/ x [ y  /  x ] ph )
65sps 1920 . 2  |-  ( A. x F/ y ph  ->  F/ x [ y  /  x ] ph )
7 sbequ12 2051 . . 3  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
87a1i 11 . 2  |-  ( A. x F/ y ph  ->  ( x  =  y  -> 
( ph  <->  [ y  /  x ] ph ) ) )
91, 3, 4, 6, 8cbv2 2078 1  |-  ( A. x F/ y ph  ->  ( A. x ph  <->  A. y [ y  /  x ] ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> 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-11 1896  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:  wl-sb8et  31845  wl-sbhbt  31846
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