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Theorem sbexyz 1876
 Description: Move existential quantifier in and out of substitution. Identical to sbex 1877 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]xφx[z / y]φ)
Distinct variable group:   x,y,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem sbexyz
StepHypRef Expression
1 sb5 1764 . . 3 ([z / y]xφy(y = z xφ))
2 exdistr 1784 . . 3 (yx(y = z φ) ↔ y(y = z xφ))
3 excom 1551 . . 3 (yx(y = z φ) ↔ xy(y = z φ))
41, 2, 33bitr2i 197 . 2 ([z / y]xφxy(y = z φ))
5 sb5 1764 . . 3 ([z / y]φy(y = z φ))
65exbii 1493 . 2 (x[z / y]φxy(y = z φ))
74, 6bitr4i 176 1 ([z / y]xφx[z / y]φ)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  ∃wex 1378  [wsb 1642 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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424 This theorem depends on definitions:  df-bi 110  df-sb 1643 This theorem is referenced by:  sbex  1877
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