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Theorem moanim 2100
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 590 . . . . 5 ((𝜑 ∧ (𝜓 ∧ [𝑦 / 𝑥]𝜓)) ↔ ((𝜑𝜓) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜓)))
21imbi1i 238 . . . 4 (((𝜑 ∧ (𝜓 ∧ [𝑦 / 𝑥]𝜓)) → 𝑥 = 𝑦) ↔ (((𝜑𝜓) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜓)) → 𝑥 = 𝑦))
3 impexp 263 . . . 4 (((𝜑 ∧ (𝜓 ∧ [𝑦 / 𝑥]𝜓)) → 𝑥 = 𝑦) ↔ (𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
4 sban 1955 . . . . . . 7 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
5 moanim.1 . . . . . . . . 9 𝑥𝜑
65sbf 1777 . . . . . . . 8 ([𝑦 / 𝑥]𝜑𝜑)
76anbi1i 458 . . . . . . 7 (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ (𝜑 ∧ [𝑦 / 𝑥]𝜓))
84, 7bitr2i 185 . . . . . 6 ((𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ [𝑦 / 𝑥](𝜑𝜓))
98anbi2i 457 . . . . 5 (((𝜑𝜓) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜓)) ↔ ((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓)))
109imbi1i 238 . . . 4 ((((𝜑𝜓) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜓)) → 𝑥 = 𝑦) ↔ (((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓)) → 𝑥 = 𝑦))
112, 3, 103bitr3i 210 . . 3 ((𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) ↔ (((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓)) → 𝑥 = 𝑦))
12112albii 1471 . 2 (∀𝑥𝑦(𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) ↔ ∀𝑥𝑦(((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓)) → 𝑥 = 𝑦))
13519.21 1583 . . 3 (∀𝑥(𝜑 → ∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) ↔ (𝜑 → ∀𝑥𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
14 19.21v 1873 . . . 4 (∀𝑦(𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) ↔ (𝜑 → ∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
1514albii 1470 . . 3 (∀𝑥𝑦(𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) ↔ ∀𝑥(𝜑 → ∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
16 ax-17 1526 . . . . 5 (𝜓 → ∀𝑦𝜓)
1716mo3h 2079 . . . 4 (∃*𝑥𝜓 ↔ ∀𝑥𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦))
1817imbi2i 226 . . 3 ((𝜑 → ∃*𝑥𝜓) ↔ (𝜑 → ∀𝑥𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
1913, 15, 183bitr4ri 213 . 2 ((𝜑 → ∃*𝑥𝜓) ↔ ∀𝑥𝑦(𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
20 ax-17 1526 . . 3 ((𝜑𝜓) → ∀𝑦(𝜑𝜓))
2120mo3h 2079 . 2 (∃*𝑥(𝜑𝜓) ↔ ∀𝑥𝑦(((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓)) → 𝑥 = 𝑦))
2212, 19, 213bitr4ri 213 1 (∃*𝑥(𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1351  wnf 1460  [wsb 1762  ∃*wmo 2027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030
This theorem is referenced by:  moanimv  2101  moaneu  2102  moanmo  2103
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