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Theorem sbal 2066
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 1990 . . . . 5  |-  ( A. x  x  =  z  ->  ( ph  <->  A. x ph ) )
21sbimi 1633 . . . 4  |-  ( [ z  /  y ] A. x  x  =  z  ->  [ z  /  y ] (
ph 
<-> 
A. x ph )
)
3 sbequ5 1971 . . . 4  |-  ( [ z  /  y ] A. x  x  =  z  <->  A. x  x  =  z )
4 sbbi 2011 . . . 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 1990 . . 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 2065 . 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 1527   [wsb 1629
This theorem is referenced by:  sbex  2067  sbalv  2068  sbcal  3038  sbcalg  3039  mo5f  23143
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630
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