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

Theorem moim 1964
 Description: "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.)
Assertion
Ref Expression
moim (∀𝑥(𝜑𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))

Proof of Theorem moim
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfa1 1434 . . 3 𝑥𝑥(𝜑𝜓)
2 ax-4 1400 . . . . . 6 (∀𝑥(𝜑𝜓) → (𝜑𝜓))
3 spsbim 1724 . . . . . 6 (∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
42, 3anim12d 318 . . . . 5 (∀𝑥(𝜑𝜓) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜓 ∧ [𝑦 / 𝑥]𝜓)))
54imim1d 69 . . . 4 (∀𝑥(𝜑𝜓) → (((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
65alimdv 1759 . . 3 (∀𝑥(𝜑𝜓) → (∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦) → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
71, 6alimd 1414 . 2 (∀𝑥(𝜑𝜓) → (∀𝑥𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦) → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
8 ax-17 1419 . . 3 (𝜓 → ∀𝑦𝜓)
98mo3h 1953 . 2 (∃*𝑥𝜓 ↔ ∀𝑥𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦))
10 ax-17 1419 . . 3 (𝜑 → ∀𝑦𝜑)
1110mo3h 1953 . 2 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
127, 9, 113imtr4g 194 1 (∀𝑥(𝜑𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))
 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:  moimi  1965  euimmo  1967  moexexdc  1984  euexex  1985  rmoim  2740  rmoimi2  2742  disjss1  3751  reusv1  4190  funmo  4917
 Copyright terms: Public domain W3C validator