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

Theorem sbex 1929
Description: Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.)
Assertion
Ref Expression
sbex ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
Distinct variable groups:   𝑥,𝑦   𝑥,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sbex
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 sbexyz 1928 . . . 4 ([𝑤 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑤 / 𝑦]𝜑)
21sbbii 1696 . . 3 ([𝑧 / 𝑤][𝑤 / 𝑦]∃𝑥𝜑 ↔ [𝑧 / 𝑤]∃𝑥[𝑤 / 𝑦]𝜑)
3 sbexyz 1928 . . 3 ([𝑧 / 𝑤]∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑)
42, 3bitri 183 . 2 ([𝑧 / 𝑤][𝑤 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑)
5 ax-17 1465 . . 3 (∃𝑥𝜑 → ∀𝑤𝑥𝜑)
65sbco2v 1870 . 2 ([𝑧 / 𝑤][𝑤 / 𝑦]∃𝑥𝜑 ↔ [𝑧 / 𝑦]∃𝑥𝜑)
7 ax-17 1465 . . . 4 (𝜑 → ∀𝑤𝜑)
87sbco2v 1870 . . 3 ([𝑧 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑)
98exbii 1542 . 2 (∃𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
104, 6, 93bitr3i 209 1 ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
Colors of variables: wff set class
Syntax hints:  wb 104  wex 1427  [wsb 1693
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-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474
This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694
This theorem is referenced by:  sbabel  2255  sbcex2  2893  sbcexg  2894
  Copyright terms: Public domain W3C validator