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Theorem moanim 1974
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 524 . . . . 5 ((𝜑 ∧ (𝜓 ∧ [𝑦 / 𝑥]𝜓)) ↔ ((𝜑𝜓) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜓)))
21imbi1i 227 . . . 4 (((𝜑 ∧ (𝜓 ∧ [𝑦 / 𝑥]𝜓)) → 𝑥 = 𝑦) ↔ (((𝜑𝜓) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜓)) → 𝑥 = 𝑦))
3 impexp 250 . . . 4 (((𝜑 ∧ (𝜓 ∧ [𝑦 / 𝑥]𝜓)) → 𝑥 = 𝑦) ↔ (𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
4 sban 1829 . . . . . . 7 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
5 moanim.1 . . . . . . . . 9 𝑥𝜑
65sbf 1660 . . . . . . . 8 ([𝑦 / 𝑥]𝜑𝜑)
76anbi1i 431 . . . . . . 7 (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ (𝜑 ∧ [𝑦 / 𝑥]𝜓))
84, 7bitr2i 174 . . . . . 6 ((𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ [𝑦 / 𝑥](𝜑𝜓))
98anbi2i 430 . . . . 5 (((𝜑𝜓) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜓)) ↔ ((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓)))
109imbi1i 227 . . . 4 ((((𝜑𝜓) ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜓)) → 𝑥 = 𝑦) ↔ (((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓)) → 𝑥 = 𝑦))
112, 3, 103bitr3i 199 . . 3 ((𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) ↔ (((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓)) → 𝑥 = 𝑦))
12112albii 1360 . 2 (∀𝑥𝑦(𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) ↔ ∀𝑥𝑦(((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓)) → 𝑥 = 𝑦))
13519.21 1475 . . 3 (∀𝑥(𝜑 → ∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) ↔ (𝜑 → ∀𝑥𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
14 19.21v 1753 . . . 4 (∀𝑦(𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) ↔ (𝜑 → ∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
1514albii 1359 . . 3 (∀𝑥𝑦(𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)) ↔ ∀𝑥(𝜑 → ∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
16 ax-17 1419 . . . . 5 (𝜓 → ∀𝑦𝜓)
1716mo3h 1953 . . . 4 (∃*𝑥𝜓 ↔ ∀𝑥𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦))
1817imbi2i 215 . . 3 ((𝜑 → ∃*𝑥𝜓) ↔ (𝜑 → ∀𝑥𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
1913, 15, 183bitr4ri 202 . 2 ((𝜑 → ∃*𝑥𝜓) ↔ ∀𝑥𝑦(𝜑 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
20 ax-17 1419 . . 3 ((𝜑𝜓) → ∀𝑦(𝜑𝜓))
2120mo3h 1953 . 2 (∃*𝑥(𝜑𝜓) ↔ ∀𝑥𝑦(((𝜑𝜓) ∧ [𝑦 / 𝑥](𝜑𝜓)) → 𝑥 = 𝑦))
2212, 19, 213bitr4ri 202 1 (∃*𝑥(𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  wal 1241  wnf 1349  [wsb 1645  ∃*wmo 1901
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904
This theorem is referenced by:  moanimv  1975  moaneu  1976  moanmo  1977
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