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Theorem sbex 1923
Description: Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.)
Assertion
Ref Expression
sbex  |-  ( [ z  /  y ] E. x ph  <->  E. x [ z  /  y ] ph )
Distinct variable groups:    x, y    x, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem sbex
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 sbexyz 1922 . . . 4  |-  ( [ w  /  y ] E. x ph  <->  E. x [ w  /  y ] ph )
21sbbii 1690 . . 3  |-  ( [ z  /  w ] [ w  /  y ] E. x ph  <->  [ z  /  w ] E. x [ w  /  y ] ph )
3 sbexyz 1922 . . 3  |-  ( [ z  /  w ] E. x [ w  / 
y ] ph  <->  E. x [ z  /  w ] [ w  /  y ] ph )
42, 3bitri 182 . 2  |-  ( [ z  /  w ] [ w  /  y ] E. x ph  <->  E. x [ z  /  w ] [ w  /  y ] ph )
5 ax-17 1460 . . 3  |-  ( E. x ph  ->  A. w E. x ph )
65sbco2v 1864 . 2  |-  ( [ z  /  w ] [ w  /  y ] E. x ph  <->  [ z  /  y ] E. x ph )
7 ax-17 1460 . . . 4  |-  ( ph  ->  A. w ph )
87sbco2v 1864 . . 3  |-  ( [ z  /  w ] [ w  /  y ] ph  <->  [ z  /  y ] ph )
98exbii 1537 . 2  |-  ( E. x [ z  /  w ] [ w  / 
y ] ph  <->  E. x [ z  /  y ] ph )
104, 6, 93bitr3i 208 1  |-  ( [ z  /  y ] E. x ph  <->  E. x [ z  /  y ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   E.wex 1422   [wsb 1687
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688
This theorem is referenced by:  sbabel  2248  sbcex2  2878  sbcexg  2879
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