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Theorem sbexyz 1921
Description: Move existential quantifier in and out of substitution. Identical to sbex 1922 except that it has an additional distinct variable constraint on  y and  z. (Contributed by Jim Kingdon, 29-Dec-2017.)
Assertion
Ref Expression
sbexyz  |-  ( [ z  /  y ] E. x ph  <->  E. x [ z  /  y ] ph )
Distinct variable group:    x, y, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem sbexyz
StepHypRef Expression
1 sb5 1809 . . 3  |-  ( [ z  /  y ] E. x ph  <->  E. y
( y  =  z  /\  E. x ph ) )
2 exdistr 1829 . . 3  |-  ( E. y E. x ( y  =  z  /\  ph )  <->  E. y ( y  =  z  /\  E. x ph ) )
3 excom 1595 . . 3  |-  ( E. y E. x ( y  =  z  /\  ph )  <->  E. x E. y
( y  =  z  /\  ph ) )
41, 2, 33bitr2i 206 . 2  |-  ( [ z  /  y ] E. x ph  <->  E. x E. y ( y  =  z  /\  ph )
)
5 sb5 1809 . . 3  |-  ( [ z  /  y ]
ph 
<->  E. y ( y  =  z  /\  ph ) )
65exbii 1537 . 2  |-  ( E. x [ z  / 
y ] ph  <->  E. x E. y ( y  =  z  /\  ph )
)
74, 6bitr4i 185 1  |-  ( [ z  /  y ] E. x ph  <->  E. x [ z  /  y ] ph )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103   E.wex 1422   [wsb 1686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-sb 1687
This theorem is referenced by:  sbex  1922
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