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Theorem sbex 1880
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 1879 . . . 4  |-  ( [ w  /  y ] E. x ph  <->  E. x [ w  /  y ] ph )
21sbbii 1648 . . 3  |-  ( [ z  /  w ] [ w  /  y ] E. x ph  <->  [ z  /  w ] E. x [ w  /  y ] ph )
3 sbexyz 1879 . . 3  |-  ( [ z  /  w ] E. x [ w  / 
y ] ph  <->  E. x [ z  /  w ] [ w  /  y ] ph )
42, 3bitri 173 . 2  |-  ( [ z  /  w ] [ w  /  y ] E. x ph  <->  E. x [ z  /  w ] [ w  /  y ] ph )
5 ax-17 1419 . . 3  |-  ( E. x ph  ->  A. w E. x ph )
65sbco2v 1821 . 2  |-  ( [ z  /  w ] [ w  /  y ] E. x ph  <->  [ z  /  y ] E. x ph )
7 ax-17 1419 . . . 4  |-  ( ph  ->  A. w ph )
87sbco2v 1821 . . 3  |-  ( [ z  /  w ] [ w  /  y ] ph  <->  [ z  /  y ] ph )
98exbii 1496 . 2  |-  ( E. x [ z  /  w ] [ w  / 
y ] ph  <->  E. x [ z  /  y ] ph )
104, 6, 93bitr3i 199 1  |-  ( [ z  /  y ] E. x ph  <->  E. x [ z  /  y ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 98   E.wex 1381   [wsb 1645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646
This theorem is referenced by:  sbabel  2203  sbcex2  2812  sbcexg  2813
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