MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbex Structured version   Visualization version   GIF version

Theorem sbex 2611
Description: Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.)
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
StepHypRef Expression
1 sbn 2538 . . 3 ([𝑧 / 𝑦] ¬ ∀𝑥 ¬ 𝜑 ↔ ¬ [𝑧 / 𝑦]∀𝑥 ¬ 𝜑)
2 sbal 2610 . . . 4 ([𝑧 / 𝑦]∀𝑥 ¬ 𝜑 ↔ ∀𝑥[𝑧 / 𝑦] ¬ 𝜑)
3 sbn 2538 . . . . 5 ([𝑧 / 𝑦] ¬ 𝜑 ↔ ¬ [𝑧 / 𝑦]𝜑)
43albii 1895 . . . 4 (∀𝑥[𝑧 / 𝑦] ¬ 𝜑 ↔ ∀𝑥 ¬ [𝑧 / 𝑦]𝜑)
52, 4bitri 264 . . 3 ([𝑧 / 𝑦]∀𝑥 ¬ 𝜑 ↔ ∀𝑥 ¬ [𝑧 / 𝑦]𝜑)
61, 5xchbinx 323 . 2 ([𝑧 / 𝑦] ¬ ∀𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥 ¬ [𝑧 / 𝑦]𝜑)
7 df-ex 1853 . . 3 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
87sbbii 2056 . 2 ([𝑧 / 𝑦]∃𝑥𝜑 ↔ [𝑧 / 𝑦] ¬ ∀𝑥 ¬ 𝜑)
9 df-ex 1853 . 2 (∃𝑥[𝑧 / 𝑦]𝜑 ↔ ¬ ∀𝑥 ¬ [𝑧 / 𝑦]𝜑)
106, 8, 93bitr4i 292 1 ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wal 1629  wex 1852  [wsb 2049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050
This theorem is referenced by:  sbmo  2664  sbabel  2942  sbcex2  3638  sbcexgOLD  39278
  Copyright terms: Public domain W3C validator