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Theorem sbex 1957
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 1956 . . . 4  |-  ( [ w  /  y ] E. x ph  <->  E. x [ w  /  y ] ph )
21sbbii 1723 . . 3  |-  ( [ z  /  w ] [ w  /  y ] E. x ph  <->  [ z  /  w ] E. x [ w  /  y ] ph )
3 sbexyz 1956 . . 3  |-  ( [ z  /  w ] E. x [ w  / 
y ] ph  <->  E. x [ z  /  w ] [ w  /  y ] ph )
42, 3bitri 183 . 2  |-  ( [ z  /  w ] [ w  /  y ] E. x ph  <->  E. x [ z  /  w ] [ w  /  y ] ph )
5 ax-17 1491 . . 3  |-  ( E. x ph  ->  A. w E. x ph )
65sbco2v 1898 . 2  |-  ( [ z  /  w ] [ w  /  y ] E. x ph  <->  [ z  /  y ] E. x ph )
7 ax-17 1491 . . . 4  |-  ( ph  ->  A. w ph )
87sbco2v 1898 . . 3  |-  ( [ z  /  w ] [ w  /  y ] ph  <->  [ z  /  y ] ph )
98exbii 1569 . 2  |-  ( E. x [ z  /  w ] [ w  / 
y ] ph  <->  E. x [ z  /  y ] ph )
104, 6, 93bitr3i 209 1  |-  ( [ z  /  y ] E. x ph  <->  E. x [ z  /  y ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   E.wex 1453   [wsb 1720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721
This theorem is referenced by:  sbabel  2284  sbcex2  2934  sbcexg  2935
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