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Theorem sbexyz 1879
 Description: Move existential quantifier in and out of substitution. Identical to sbex 1880 except that it has an additional distinct variable constraint on and . (Contributed by Jim Kingdon, 29-Dec-2017.)
Assertion
Ref Expression
sbexyz
Distinct variable group:   ,,
Allowed substitution hints:   (,,)

Proof of Theorem sbexyz
StepHypRef Expression
1 sb5 1767 . . 3
2 exdistr 1787 . . 3
3 excom 1554 . . 3
41, 2, 33bitr2i 197 . 2
5 sb5 1767 . . 3
65exbii 1496 . 2
74, 6bitr4i 176 1
 Colors of variables: wff set class Syntax hints:   wa 97   wb 98  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-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-sb 1646 This theorem is referenced by:  sbex  1880
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