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Theorem sbexyz 1939
Description: Move existential quantifier in and out of substitution. Identical to sbex 1940 except that it has an additional disjoint variable condition on  y ,  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 1826 . . 3  |-  ( [ z  /  y ] E. x ph  <->  E. y
( y  =  z  /\  E. x ph ) )
2 exdistr 1846 . . 3  |-  ( E. y E. x ( y  =  z  /\  ph )  <->  E. y ( y  =  z  /\  E. x ph ) )
3 excom 1610 . . 3  |-  ( E. y E. x ( y  =  z  /\  ph )  <->  E. x E. y
( y  =  z  /\  ph ) )
41, 2, 33bitr2i 207 . 2  |-  ( [ z  /  y ] E. x ph  <->  E. x E. y ( y  =  z  /\  ph )
)
5 sb5 1826 . . 3  |-  ( [ z  /  y ]
ph 
<->  E. y ( y  =  z  /\  ph ) )
65exbii 1552 . 2  |-  ( E. x [ z  / 
y ] ph  <->  E. x E. y ( y  =  z  /\  ph )
)
74, 6bitr4i 186 1  |-  ( [ z  /  y ] E. x ph  <->  E. x [ z  /  y ] ph )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104   E.wex 1436   [wsb 1703
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-11 1452  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482
This theorem depends on definitions:  df-bi 116  df-sb 1704
This theorem is referenced by:  sbex  1940
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