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Theorem moimv 2014
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 1694 . . . . . . 7 ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜓 → (𝜑𝜓)))
4 nfv 1466 . . . . . . . 8 𝑥𝜑
54sbf 1707 . . . . . . 7 ([𝑦 / 𝑥]𝜑𝜑)
6 sbim 1875 . . . . . . 7 ([𝑦 / 𝑥](𝜓 → (𝜑𝜓)) ↔ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥](𝜑𝜓)))
73, 5, 63imtr3i 198 . . . . . 6 (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥](𝜑𝜓)))
82, 7anim12d 328 . . . . 5 (𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → ((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓))))
98imim1d 74 . . . 4 (𝜑 → ((((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓)) → 𝑥 = 𝑦) → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
1092alimdv 1809 . . 3 (𝜑 → (∀𝑥𝑦(((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓)) → 𝑥 = 𝑦) → ∀𝑥𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
11 ax-17 1464 . . . 4 ((𝜑𝜓) → ∀𝑦(𝜑𝜓))
1211mo3h 2001 . . 3 (∃*𝑥(𝜑𝜓) ↔ ∀𝑥𝑦(((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓)) → 𝑥 = 𝑦))
13 ax-17 1464 . . . 4 (𝜓 → ∀𝑦𝜓)
1413mo3h 2001 . . 3 (∃*𝑥𝜓 ↔ ∀𝑥𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦))
1510, 12, 143imtr4g 203 . 2 (𝜑 → (∃*𝑥(𝜑𝜓) → ∃*𝑥𝜓))
1615com12 30 1 (∃*𝑥(𝜑𝜓) → (𝜑 → ∃*𝑥𝜓))
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: (None)
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