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

Theorem moimv 1966
Description: Move antecedent outside of "at most one." (Contributed by NM, 28-Jul-1995.)
Assertion
Ref Expression
moimv (∃*𝑥(𝜑𝜓) → (𝜑 → ∃*𝑥𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem moimv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ax-1 5 . . . . . . 7 (𝜓 → (𝜑𝜓))
21a1i 9 . . . . . 6 (𝜑 → (𝜓 → (𝜑𝜓)))
32sbimi 1647 . . . . . . 7 ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜓 → (𝜑𝜓)))
4 nfv 1421 . . . . . . . 8 𝑥𝜑
54sbf 1660 . . . . . . 7 ([𝑦 / 𝑥]𝜑𝜑)
6 sbim 1827 . . . . . . 7 ([𝑦 / 𝑥](𝜓 → (𝜑𝜓)) ↔ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥](𝜑𝜓)))
73, 5, 63imtr3i 189 . . . . . 6 (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥](𝜑𝜓)))
82, 7anim12d 318 . . . . 5 (𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → ((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓))))
98imim1d 69 . . . 4 (𝜑 → ((((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓)) → 𝑥 = 𝑦) → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
1092alimdv 1761 . . 3 (𝜑 → (∀𝑥𝑦(((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓)) → 𝑥 = 𝑦) → ∀𝑥𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
11 ax-17 1419 . . . 4 ((𝜑𝜓) → ∀𝑦(𝜑𝜓))
1211mo3h 1953 . . 3 (∃*𝑥(𝜑𝜓) ↔ ∀𝑥𝑦(((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓)) → 𝑥 = 𝑦))
13 ax-17 1419 . . . 4 (𝜓 → ∀𝑦𝜓)
1413mo3h 1953 . . 3 (∃*𝑥𝜓 ↔ ∀𝑥𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦))
1510, 12, 143imtr4g 194 . 2 (𝜑 → (∃*𝑥(𝜑𝜓) → ∃*𝑥𝜓))
1615com12 27 1 (∃*𝑥(𝜑𝜓) → (𝜑 → ∃*𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wal 1241  [wsb 1645  ∃*wmo 1901
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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator