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Theorem sbex 2016
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 2015 . . . 4  |-  ( [ w  /  y ] E. x ph  <->  E. x [ w  /  y ] ph )
21sbbii 1776 . . 3  |-  ( [ z  /  w ] [ w  /  y ] E. x ph  <->  [ z  /  w ] E. x [ w  /  y ] ph )
3 sbexyz 2015 . . 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 1537 . . 3  |-  ( E. x ph  ->  A. w E. x ph )
65sbco2vh 1957 . 2  |-  ( [ z  /  w ] [ w  /  y ] E. x ph  <->  [ z  /  y ] E. x ph )
7 ax-17 1537 . . . 4  |-  ( ph  ->  A. w ph )
87sbco2vh 1957 . . 3  |-  ( [ z  /  w ] [ w  /  y ] ph  <->  [ z  /  y ] ph )
98exbii 1616 . 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 1503   [wsb 1773
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774
This theorem is referenced by:  sbabel  2359  sbcex2  3031  sbcexg  3032
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