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Theorem sbex 2004
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 2003 . . . 4  |-  ( [ w  /  y ] E. x ph  <->  E. x [ w  /  y ] ph )
21sbbii 1765 . . 3  |-  ( [ z  /  w ] [ w  /  y ] E. x ph  <->  [ z  /  w ] E. x [ w  /  y ] ph )
3 sbexyz 2003 . . 3  |-  ( [ z  /  w ] E. x [ w  / 
y ] ph  <->  E. x [ z  /  w ] [ w  /  y ] ph )
42, 3bitri 184 . 2  |-  ( [ z  /  w ] [ w  /  y ] E. x ph  <->  E. x [ z  /  w ] [ w  /  y ] ph )
5 ax-17 1526 . . 3  |-  ( E. x ph  ->  A. w E. x ph )
65sbco2vh 1945 . 2  |-  ( [ z  /  w ] [ w  /  y ] E. x ph  <->  [ z  /  y ] E. x ph )
7 ax-17 1526 . . . 4  |-  ( ph  ->  A. w ph )
87sbco2vh 1945 . . 3  |-  ( [ z  /  w ] [ w  /  y ] ph  <->  [ z  /  y ] ph )
98exbii 1605 . 2  |-  ( E. x [ z  /  w ] [ w  / 
y ] ph  <->  E. x [ z  /  y ] ph )
104, 6, 93bitr3i 210 1  |-  ( [ z  /  y ] E. x ph  <->  E. x [ z  /  y ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 105   E.wex 1492   [wsb 1762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763
This theorem is referenced by:  sbabel  2346  sbcex2  3016  sbcexg  3017
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