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Theorem sbalOLD 2175
Description: Obsolete proof of sbal 2174 as of 29-Sep-2018. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sbalOLD  |-  ( [ 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 sbalOLD
StepHypRef Expression
1 ax16gb 2015 . . . . 5  |-  ( A. x  x  =  z  ->  ( ph  <->  A. x ph ) )
21sbimi 1706 . . . 4  |-  ( [ z  /  y ] A. x  x  =  z  ->  [ z  /  y ] (
ph 
<-> 
A. x ph )
)
3 sbequ5 2078 . . . 4  |-  ( [ z  /  y ] A. x  x  =  z  <->  A. x  x  =  z )
4 sbbi 2093 . . . 4  |-  ( [ z  /  y ] ( ph  <->  A. x ph )  <->  ( [ z  /  y ] ph  <->  [ z  /  y ] A. x ph )
)
52, 3, 43imtr3i 265 . . 3  |-  ( A. x  x  =  z  ->  ( [ z  / 
y ] ph  <->  [ z  /  y ] A. x ph ) )
6 ax16gb 2015 . . 3  |-  ( A. x  x  =  z  ->  ( [ z  / 
y ] ph  <->  A. x [ z  /  y ] ph ) )
75, 6bitr3d 255 . 2  |-  ( A. x  x  =  z  ->  ( [ z  / 
y ] A. x ph 
<-> 
A. x [ z  /  y ] ph ) )
8 sbal1 2171 . 2  |-  ( -. 
A. x  x  =  z  ->  ( [
z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph ) )
97, 8pm2.61i 164 1  |-  ( [ z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   A.wal 1367   [wsb 1700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1587  df-nf 1590  df-sb 1701
This theorem is referenced by: (None)
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