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Theorem wl-sbnf1 31927
Description: Two ways expressing that  x is effectively not free in  ph. Simplified version of sbnf2 2278. Note: This theorem shows that sbnf2 2278 has unnecessary distinct variable constraints. (Contributed by Wolf Lammen, 28-Jul-2019.)
Assertion
Ref Expression
wl-sbnf1  |-  ( A. x F/ y ph  ->  ( F/ x ph  <->  A. x A. y ( ph  ->  [ y  /  x ] ph ) ) )

Proof of Theorem wl-sbnf1
StepHypRef Expression
1 df-nf 1678 . 2  |-  ( F/ x ph  <->  A. x
( ph  ->  A. x ph ) )
2 nfa1 1989 . . 3  |-  F/ x A. x F/ y ph
3 wl-sbhbt 31926 . . 3  |-  ( A. x F/ y ph  ->  ( ( ph  ->  A. x ph )  <->  A. y ( ph  ->  [ y  /  x ] ph ) ) )
42, 3albid 1973 . 2  |-  ( A. x F/ y ph  ->  ( A. x ( ph  ->  A. x ph )  <->  A. x A. y (
ph  ->  [ y  /  x ] ph ) ) )
51, 4syl5bb 265 1  |-  ( A. x F/ y ph  ->  ( F/ x ph  <->  A. x A. y ( ph  ->  [ y  /  x ] ph ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> 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: (None)
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