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Theorem sbexyz 2015
Description: Move existential quantifier in and out of substitution. Identical to sbex 2016 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]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
Distinct variable group:   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sbexyz
StepHypRef Expression
1 sb5 1899 . . 3 ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑦(𝑦 = 𝑧 ∧ ∃𝑥𝜑))
2 exdistr 1921 . . 3 (∃𝑦𝑥(𝑦 = 𝑧𝜑) ↔ ∃𝑦(𝑦 = 𝑧 ∧ ∃𝑥𝜑))
3 excom 1675 . . 3 (∃𝑦𝑥(𝑦 = 𝑧𝜑) ↔ ∃𝑥𝑦(𝑦 = 𝑧𝜑))
41, 2, 33bitr2i 208 . 2 ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥𝑦(𝑦 = 𝑧𝜑))
5 sb5 1899 . . 3 ([𝑧 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝑧𝜑))
65exbii 1616 . 2 (∃𝑥[𝑧 / 𝑦]𝜑 ↔ ∃𝑥𝑦(𝑦 = 𝑧𝜑))
74, 6bitr4i 187 1 ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wex 1503  [wsb 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545
This theorem depends on definitions:  df-bi 117  df-sb 1774
This theorem is referenced by:  sbex  2016
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