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Theorem bj-sbnf 32650
Description: Move non-free predicate in and out of substitution; see sbal 2181 and sbex 2183. (Contributed by BJ, 2-May-2019.)
Assertion
Ref Expression
bj-sbnf  |-  ( [ z  /  y ] F/ x ph  <->  F/ x [ z  /  y ] ph )
Distinct variable groups:    x, y    x, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem bj-sbnf
StepHypRef Expression
1 sbim 2094 . . . 4  |-  ( [ z  /  y ] ( ph  ->  A. x ph )  <->  ( [ z  /  y ] ph  ->  [ z  /  y ] A. x ph )
)
2 sbal 2181 . . . . 5  |-  ( [ z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
32imbi2i 312 . . . 4  |-  ( ( [ z  /  y ] ph  ->  [ z  /  y ] A. x ph )  <->  ( [
z  /  y ]
ph  ->  A. x [ z  /  y ] ph ) )
41, 3bitri 249 . . 3  |-  ( [ z  /  y ] ( ph  ->  A. x ph )  <->  ( [ z  /  y ] ph  ->  A. x [ z  /  y ] ph ) )
54albii 1611 . 2  |-  ( A. x [ z  /  y ] ( ph  ->  A. x ph )  <->  A. x
( [ z  / 
y ] ph  ->  A. x [ z  / 
y ] ph )
)
6 df-nf 1591 . . . 4  |-  ( F/ x ph  <->  A. x
( ph  ->  A. x ph ) )
76sbbii 1709 . . 3  |-  ( [ z  /  y ] F/ x ph  <->  [ z  /  y ] A. x ( ph  ->  A. x ph ) )
8 sbal 2181 . . 3  |-  ( [ z  /  y ] A. x ( ph  ->  A. x ph )  <->  A. x [ z  / 
y ] ( ph  ->  A. x ph )
)
97, 8bitri 249 . 2  |-  ( [ z  /  y ] F/ x ph  <->  A. x [ z  /  y ] ( ph  ->  A. x ph ) )
10 df-nf 1591 . 2  |-  ( F/ x [ z  / 
y ] ph  <->  A. x
( [ z  / 
y ] ph  ->  A. x [ z  / 
y ] ph )
)
115, 9, 103bitr4i 277 1  |-  ( [ z  /  y ] F/ x ph  <->  F/ x [ z  /  y ] ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1368   F/wnf 1590   [wsb 1702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1588  df-nf 1591  df-sb 1703
This theorem is referenced by:  bj-nfcf  32730
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