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Theorem moanim 2017
Description: Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.)
Hypothesis
Ref Expression
moanim.1 𝑥𝜑
Assertion
Ref Expression
moanim (∃*𝑥(𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝜓))

Proof of Theorem moanim
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 anandi 555 . . . . 5 ((𝜑 ∧ (𝜓 ∧ [𝑦 / 𝑥]𝜓)) ↔ ((𝜑𝜓) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜓)))
21imbi1i 236 . . . 4 (((𝜑 ∧ (𝜓 ∧ [𝑦 / 𝑥]𝜓)) → 𝑥 = 𝑦) ↔ (((𝜑𝜓) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜓)) → 𝑥 = 𝑦))
3 impexp 259 . . . 4 (((𝜑 ∧ (𝜓 ∧ [𝑦 / 𝑥]𝜓)) → 𝑥 = 𝑦) ↔ (𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
4 sban 1872 . . . . . . 7 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
5 moanim.1 . . . . . . . . 9 𝑥𝜑
65sbf 1702 . . . . . . . 8 ([𝑦 / 𝑥]𝜑𝜑)
76anbi1i 446 . . . . . . 7 (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ (𝜑 ∧ [𝑦 / 𝑥]𝜓))
84, 7bitr2i 183 . . . . . 6 ((𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ [𝑦 / 𝑥](𝜑𝜓))
98anbi2i 445 . . . . 5 (((𝜑𝜓) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜓)) ↔ ((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓)))
109imbi1i 236 . . . 4 ((((𝜑𝜓) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜓)) → 𝑥 = 𝑦) ↔ (((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓)) → 𝑥 = 𝑦))
112, 3, 103bitr3i 208 . . 3 ((𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) ↔ (((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓)) → 𝑥 = 𝑦))
12112albii 1401 . 2 (∀𝑥𝑦(𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) ↔ ∀𝑥𝑦(((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓)) → 𝑥 = 𝑦))
13519.21 1516 . . 3 (∀𝑥(𝜑 → ∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) ↔ (𝜑 → ∀𝑥𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
14 19.21v 1796 . . . 4 (∀𝑦(𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) ↔ (𝜑 → ∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
1514albii 1400 . . 3 (∀𝑥𝑦(𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) ↔ ∀𝑥(𝜑 → ∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
16 ax-17 1460 . . . . 5 (𝜓 → ∀𝑦𝜓)
1716mo3h 1996 . . . 4 (∃*𝑥𝜓 ↔ ∀𝑥𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦))
1817imbi2i 224 . . 3 ((𝜑 → ∃*𝑥𝜓) ↔ (𝜑 → ∀𝑥𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
1913, 15, 183bitr4ri 211 . 2 ((𝜑 → ∃*𝑥𝜓) ↔ ∀𝑥𝑦(𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
20 ax-17 1460 . . 3 ((𝜑𝜓) → ∀𝑦(𝜑𝜓))
2120mo3h 1996 . 2 (∃*𝑥(𝜑𝜓) ↔ ∀𝑥𝑦(((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓)) → 𝑥 = 𝑦))
2212, 19, 213bitr4ri 211 1 (∃*𝑥(𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283  wnf 1390  [wsb 1687  ∃*wmo 1944
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947
This theorem is referenced by:  moanimv  2018  moaneu  2019  moanmo  2020
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