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Theorem moim 2012
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 1479 . . 3 𝑥𝑥(𝜑𝜓)
2 ax-4 1445 . . . . . 6 (∀𝑥(𝜑𝜓) → (𝜑𝜓))
3 spsbim 1771 . . . . . 6 (∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
42, 3anim12d 328 . . . . 5 (∀𝑥(𝜑𝜓) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜓 ∧ [𝑦 / 𝑥]𝜓)))
54imim1d 74 . . . 4 (∀𝑥(𝜑𝜓) → (((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
65alimdv 1807 . . 3 (∀𝑥(𝜑𝜓) → (∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦) → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
71, 6alimd 1459 . 2 (∀𝑥(𝜑𝜓) → (∀𝑥𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦) → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
8 ax-17 1464 . . 3 (𝜓 → ∀𝑦𝜓)
98mo3h 2001 . 2 (∃*𝑥𝜓 ↔ ∀𝑥𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦))
10 ax-17 1464 . . 3 (𝜑 → ∀𝑦𝜑)
1110mo3h 2001 . 2 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
127, 9, 113imtr4g 203 1 (∀𝑥(𝜑𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1287  [wsb 1692  ∃*wmo 1949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952
This theorem is referenced by:  moimi  2013  euimmo  2015  moexexdc  2032  euexex  2033  rmoim  2814  rmoimi2  2816  disjss1  3820  reusv1  4271  funmo  5017
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