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Theorem mo3h 2059
 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 1442 . . . . . 6 𝑦𝜑
32eu2 2050 . . . . 5 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
43imbi2i 225 . . . 4 ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑥𝜑 → (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
5 df-mo 2010 . . . 4 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
6 anclb 317 . . . 4 ((∃𝑥𝜑 → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (∃𝑥𝜑 → (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))))
74, 5, 63bitr4i 211 . . 3 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
8 19.38 1656 . . . . 5 ((∃𝑥𝜑 → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) → ∀𝑥(𝜑 → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
9219.21 1563 . . . . . 6 (∀𝑦(𝜑 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ (𝜑 → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
109albii 1450 . . . . 5 (∀𝑥𝑦(𝜑 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) ↔ ∀𝑥(𝜑 → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
118, 10sylibr 133 . . . 4 ((∃𝑥𝜑 → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) → ∀𝑥𝑦(𝜑 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
12 anabs5 563 . . . . . 6 ((𝜑 ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) ↔ (𝜑 ∧ [𝑦 / 𝑥]𝜑))
13 pm3.31 260 . . . . . 6 ((𝜑 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) → ((𝜑 ∧ (𝜑 ∧ [𝑦 / 𝑥]𝜑)) → 𝑥 = 𝑦))
1412, 13syl5bir 152 . . . . 5 ((𝜑 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
15142alimi 1436 . . . 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 120 . 2 (∃*𝑥𝜑 → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
183simplbi2com 1424 . . 3 (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (∃𝑥𝜑 → ∃!𝑥𝜑))
1918, 5sylibr 133 . 2 (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃*𝑥𝜑)
2017, 19impbii 125 1 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1333  ∃wex 1472  [wsb 1742  ∃!weu 2006  ∃*wmo 2007 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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515 This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010 This theorem is referenced by:  mo3  2060  mo2dc  2061  mo4f  2066  moim  2070  moimv  2072  moanim  2080  mopick  2084
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