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Theorem sbal 1924
Description: Move universal quantifier in and out of substitution. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 12-Feb-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
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 sbalyz 1923 . . . 4  |-  ( [ w  /  y ] A. x ph  <->  A. x [ w  /  y ] ph )
21sbbii 1695 . . 3  |-  ( [ z  /  w ] [ w  /  y ] A. x ph  <->  [ z  /  w ] A. x [ w  /  y ] ph )
3 sbalyz 1923 . . 3  |-  ( [ z  /  w ] A. x [ w  / 
y ] ph  <->  A. x [ z  /  w ] [ w  /  y ] ph )
42, 3bitri 182 . 2  |-  ( [ z  /  w ] [ w  /  y ] A. x ph  <->  A. x [ z  /  w ] [ w  /  y ] ph )
5 ax-17 1464 . . 3  |-  ( A. x ph  ->  A. w A. x ph )
65sbco2v 1869 . 2  |-  ( [ z  /  w ] [ w  /  y ] A. x ph  <->  [ z  /  y ] A. x ph )
7 ax-17 1464 . . . 4  |-  ( ph  ->  A. w ph )
87sbco2v 1869 . . 3  |-  ( [ z  /  w ] [ w  /  y ] ph  <->  [ z  /  y ] ph )
98albii 1404 . 2  |-  ( A. x [ z  /  w ] [ w  /  y ] ph  <->  A. x [ z  /  y ] ph )
104, 6, 93bitr3i 208 1  |-  ( [ z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   A.wal 1287   [wsb 1692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693
This theorem is referenced by:  sbal1  1926  sbalv  1929  sbcal  2890  sbcalg  2891
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