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Theorem sbal 2079
Description: Move universal quantifier in and out of substitution. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbal  |-  ( [ z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
Distinct variable groups:    x, y    x, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem sbal
StepHypRef Expression
1 a16gb 2003 . . . . 5  |-  ( A. x  x  =  z  ->  ( ph  <->  A. x ph ) )
21sbimi 1642 . . . 4  |-  ( [ z  /  y ] A. x  x  =  z  ->  [ z  /  y ] (
ph 
<-> 
A. x ph )
)
3 sbequ5 1984 . . . 4  |-  ( [ z  /  y ] A. x  x  =  z  <->  A. x  x  =  z )
4 sbbi 2024 . . . 4  |-  ( [ z  /  y ] ( ph  <->  A. x ph )  <->  ( [ z  /  y ] ph  <->  [ z  /  y ] A. x ph )
)
52, 3, 43imtr3i 256 . . 3  |-  ( A. x  x  =  z  ->  ( [ z  / 
y ] ph  <->  [ z  /  y ] A. x ph ) )
6 a16gb 2003 . . 3  |-  ( A. x  x  =  z  ->  ( [ z  / 
y ] ph  <->  A. x [ z  /  y ] ph ) )
75, 6bitr3d 246 . 2  |-  ( A. x  x  =  z  ->  ( [ z  / 
y ] A. x ph 
<-> 
A. x [ z  /  y ] ph ) )
8 sbal1 2078 . 2  |-  ( -. 
A. x  x  =  z  ->  ( [
z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph ) )
97, 8pm2.61i 156 1  |-  ( [ z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wal 1530   [wsb 1638
This theorem is referenced by:  sbex  2080  sbalv  2081  sbcal  3051  sbcalg  3052  mo5f  23159
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639
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