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Theorem sbex 2184
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 2091 . . 3  |-  ( [ z  /  y ]  -.  A. x  -.  ph  <->  -. 
[ z  /  y ] A. x  -.  ph )
2 sbal 2182 . . . 4  |-  ( [ z  /  y ] A. x  -.  ph  <->  A. x [ z  / 
y ]  -.  ph )
3 sbn 2091 . . . . 5  |-  ( [ z  /  y ]  -.  ph  <->  -.  [ z  /  y ] ph )
43albii 1611 . . . 4  |-  ( A. x [ z  /  y ]  -.  ph  <->  A. x  -.  [
z  /  y ]
ph )
52, 4bitri 249 . . 3  |-  ( [ z  /  y ] A. x  -.  ph  <->  A. x  -.  [ z  /  y ] ph )
61, 5xchbinx 310 . 2  |-  ( [ z  /  y ]  -.  A. x  -.  ph  <->  -. 
A. x  -.  [
z  /  y ]
ph )
7 df-ex 1588 . . 3  |-  ( E. x ph  <->  -.  A. x  -.  ph )
87sbbii 1709 . 2  |-  ( [ z  /  y ] E. x ph  <->  [ z  /  y ]  -.  A. x  -.  ph )
9 df-ex 1588 . 2  |-  ( E. x [ z  / 
y ] ph  <->  -.  A. x  -.  [ z  /  y ] ph )
106, 8, 93bitr4i 277 1  |-  ( [ z  /  y ] E. x ph  <->  E. x [ z  /  y ] ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184   A.wal 1368   E.wex 1587   [wsb 1702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1588  df-nf 1591  df-sb 1703
This theorem is referenced by:  sbmo  2323  sbabel  2644  sbcex2  3342  sbcexgOLD  3343
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