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Theorem mo3h 2001
Description: Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that 𝑦 not occur in 𝜑 in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) (New usage is discouraged.)
Hypothesis
Ref Expression
mo3h.1 (𝜑 → ∀𝑦𝜑)
Assertion
Ref Expression
mo3h (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem mo3h
StepHypRef Expression
1 mo3h.1 . . . . . . 7 (𝜑 → ∀𝑦𝜑)
21nfi 1396 . . . . . 6 𝑦𝜑
32eu2 1992 . . . . 5 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
43imbi2i 224 . . . 4 ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑥𝜑 → (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
5 df-mo 1952 . . . 4 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
6 anclb 312 . . . 4 ((∃𝑥𝜑 → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (∃𝑥𝜑 → (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
74, 5, 63bitr4i 210 . . 3 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
8 19.38 1611 . . . . 5 ((∃𝑥𝜑 → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) → ∀𝑥(𝜑 → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
9219.21 1520 . . . . . 6 (∀𝑦(𝜑 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝜑 → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
109albii 1404 . . . . 5 (∀𝑥𝑦(𝜑 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ ∀𝑥(𝜑 → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
118, 10sylibr 132 . . . 4 ((∃𝑥𝜑 → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) → ∀𝑥𝑦(𝜑 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
12 anabs5 540 . . . . . 6 ((𝜑 ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) ↔ (𝜑 ∧ [𝑦 / 𝑥]𝜑))
13 pm3.31 258 . . . . . 6 ((𝜑 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) → ((𝜑 ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) → 𝑥 = 𝑦))
1412, 13syl5bir 151 . . . . 5 ((𝜑 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
15142alimi 1390 . . . 4 (∀𝑥𝑦(𝜑 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
1611, 15syl 14 . . 3 ((∃𝑥𝜑 → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
177, 16sylbi 119 . 2 (∃*𝑥𝜑 → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
183simplbi2com 1378 . . 3 (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (∃𝑥𝜑 → ∃!𝑥𝜑))
1918, 5sylibr 132 . 2 (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃*𝑥𝜑)
2017, 19impbii 124 1 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1287  wex 1426  [wsb 1692  ∃!weu 1948  ∃*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:  mo3  2002  mo2dc  2003  mo4f  2008  moim  2012  moimv  2014  moanim  2022  mopick  2026
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