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Theorem sbex 2164
Description: Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.)
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
StepHypRef Expression
1 sbn 2097 . . 3  |-  ( [ z  /  y ]  -.  A. x  -.  ph  <->  -. 
[ z  /  y ] A. x  -.  ph )
2 sbal 2163 . . . 4  |-  ( [ z  /  y ] A. x  -.  ph  <->  A. x [ z  / 
y ]  -.  ph )
3 sbn 2097 . . . . 5  |-  ( [ z  /  y ]  -.  ph  <->  -.  [ z  /  y ] ph )
43albii 1572 . . . 4  |-  ( A. x [ z  /  y ]  -.  ph  <->  A. x  -.  [
z  /  y ]
ph )
52, 4bitri 241 . . 3  |-  ( [ z  /  y ] A. x  -.  ph  <->  A. x  -.  [ z  /  y ] ph )
61, 5xchbinx 302 . 2  |-  ( [ z  /  y ]  -.  A. x  -.  ph  <->  -. 
A. x  -.  [
z  /  y ]
ph )
7 df-ex 1548 . . 3  |-  ( E. x ph  <->  -.  A. x  -.  ph )
87sbbii 1660 . 2  |-  ( [ z  /  y ] E. x ph  <->  [ z  /  y ]  -.  A. x  -.  ph )
9 df-ex 1548 . 2  |-  ( E. x [ z  / 
y ] ph  <->  -.  A. x  -.  [ z  /  y ] ph )
106, 8, 93bitr4i 269 1  |-  ( [ z  /  y ] E. x ph  <->  E. x [ z  /  y ] ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177   A.wal 1546   E.wex 1547   [wsb 1655
This theorem is referenced by:  sbmo  2270  sbabel  2551  sbcex2  3155  sbcexg  3156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656
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