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Theorem sbal 2174
Description: Move universal quantifier in and out of substitution. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 29-Sep-2018.)
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 nfae 2003 . . . 4  |-  F/ y A. x  x  =  z
2 ax16gb 2015 . . . 4  |-  ( A. x  x  =  z  ->  ( ph  <->  A. x ph ) )
31, 2sbbid 2095 . . 3  |-  ( A. x  x  =  z  ->  ( [ z  / 
y ] ph  <->  [ z  /  y ] A. x ph ) )
4 ax16gb 2015 . . 3  |-  ( A. x  x  =  z  ->  ( [ z  / 
y ] ph  <->  A. x [ z  /  y ] ph ) )
53, 4bitr3d 255 . 2  |-  ( A. x  x  =  z  ->  ( [ z  / 
y ] A. x ph 
<-> 
A. x [ z  /  y ] ph ) )
6 sbal1 2171 . 2  |-  ( -. 
A. x  x  =  z  ->  ( [
z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph ) )
75, 6pm2.61i 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:  sbex  2176  sbalv  2177  sbcal  3238  sbcalgOLD  3239  ax11-pm2  32344  bj-sbnf  32347
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