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Theorem moimv 2041
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 6 . . . . . . 7 (𝜓 → (𝜑𝜓))
21a1i 9 . . . . . 6 (𝜑 → (𝜓 → (𝜑𝜓)))
32sbimi 1720 . . . . . . 7 ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜓 → (𝜑𝜓)))
4 nfv 1491 . . . . . . . 8 𝑥𝜑
54sbf 1733 . . . . . . 7 ([𝑦 / 𝑥]𝜑𝜑)
6 sbim 1902 . . . . . . 7 ([𝑦 / 𝑥](𝜓 → (𝜑𝜓)) ↔ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥](𝜑𝜓)))
73, 5, 63imtr3i 199 . . . . . 6 (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥](𝜑𝜓)))
82, 7anim12d 331 . . . . 5 (𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → ((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓))))
98imim1d 75 . . . 4 (𝜑 → ((((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓)) → 𝑥 = 𝑦) → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
1092alimdv 1835 . . 3 (𝜑 → (∀𝑥𝑦(((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓)) → 𝑥 = 𝑦) → ∀𝑥𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
11 ax-17 1489 . . . 4 ((𝜑𝜓) → ∀𝑦(𝜑𝜓))
1211mo3h 2028 . . 3 (∃*𝑥(𝜑𝜓) ↔ ∀𝑥𝑦(((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓)) → 𝑥 = 𝑦))
13 ax-17 1489 . . . 4 (𝜓 → ∀𝑦𝜓)
1413mo3h 2028 . . 3 (∃*𝑥𝜓 ↔ ∀𝑥𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦))
1510, 12, 143imtr4g 204 . 2 (𝜑 → (∃*𝑥(𝜑𝜓) → ∃*𝑥𝜓))
1615com12 30 1 (∃*𝑥(𝜑𝜓) → (𝜑 → ∃*𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1312  [wsb 1718  ∃*wmo 1976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979
This theorem is referenced by: (None)
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