Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbexyz GIF version

Theorem sbexyz 1934
 Description: Move existential quantifier in and out of substitution. Identical to sbex 1935 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 1822 . . 3 ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑦(𝑦 = 𝑧 ∧ ∃𝑥𝜑))
2 exdistr 1842 . . 3 (∃𝑦𝑥(𝑦 = 𝑧𝜑) ↔ ∃𝑦(𝑦 = 𝑧 ∧ ∃𝑥𝜑))
3 excom 1606 . . 3 (∃𝑦𝑥(𝑦 = 𝑧𝜑) ↔ ∃𝑥𝑦(𝑦 = 𝑧𝜑))
41, 2, 33bitr2i 207 . 2 ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥𝑦(𝑦 = 𝑧𝜑))
5 sb5 1822 . . 3 ([𝑧 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝑧𝜑))
65exbii 1548 . 2 (∃𝑥[𝑧 / 𝑦]𝜑 ↔ ∃𝑥𝑦(𝑦 = 𝑧𝜑))
74, 6bitr4i 186 1 ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104  ∃wex 1433  [wsb 1699 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 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-11 1449  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479 This theorem depends on definitions:  df-bi 116  df-sb 1700 This theorem is referenced by:  sbex  1935
 Copyright terms: Public domain W3C validator