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Theorem sbex 2258
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 2185 . . 3  |-  ( [ z  /  y ]  -.  A. x  -.  ph  <->  -. 
[ z  /  y ] A. x  -.  ph )
2 sbal 2257 . . . 4  |-  ( [ z  /  y ] A. x  -.  ph  <->  A. x [ z  / 
y ]  -.  ph )
3 sbn 2185 . . . . 5  |-  ( [ z  /  y ]  -.  ph  <->  -.  [ z  /  y ] ph )
43albii 1687 . . . 4  |-  ( A. x [ z  /  y ]  -.  ph  <->  A. x  -.  [
z  /  y ]
ph )
52, 4bitri 252 . . 3  |-  ( [ z  /  y ] A. x  -.  ph  <->  A. x  -.  [ z  /  y ] ph )
61, 5xchbinx 311 . 2  |-  ( [ z  /  y ]  -.  A. x  -.  ph  <->  -. 
A. x  -.  [
z  /  y ]
ph )
7 df-ex 1660 . . 3  |-  ( E. x ph  <->  -.  A. x  -.  ph )
87sbbii 1793 . 2  |-  ( [ z  /  y ] E. x ph  <->  [ z  /  y ]  -.  A. x  -.  ph )
9 df-ex 1660 . 2  |-  ( E. x [ z  / 
y ] ph  <->  -.  A. x  -.  [ z  /  y ] ph )
106, 8, 93bitr4i 280 1  |-  ( [ z  /  y ] E. x ph  <->  E. x [ z  /  y ] ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187   A.wal 1435   E.wex 1659   [wsb 1786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-ex 1660  df-nf 1664  df-sb 1787
This theorem is referenced by:  sbmo  2311  sbabel  2616  sbabelOLD  2617  sbcex2  3350  sbcexgOLD  36762
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