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Theorem modc 1959
Description: Equivalent definitions of "there exists at most one," given decidable existence. (Contributed by Jim Kingdon, 1-Jul-2018.)
Hypothesis
Ref Expression
modc.1 𝑦𝜑
Assertion
Ref Expression
modc (DECID𝑥𝜑 → (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem modc
StepHypRef Expression
1 modc.1 . . 3 𝑦𝜑
21mo23 1957 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
3 exmiddc 755 . . 3 (DECID𝑥𝜑 → (∃𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
41mor 1958 . . . 4 (∃𝑥𝜑 → (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
51mo2n 1944 . . . . 5 (¬ ∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
65a1d 22 . . . 4 (¬ ∃𝑥𝜑 → (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
74, 6jaoi 646 . . 3 ((∃𝑥𝜑 ∨ ¬ ∃𝑥𝜑) → (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
83, 7syl 14 . 2 (DECID𝑥𝜑 → (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
92, 8impbid2 135 1 (DECID𝑥𝜑 → (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wo 639  DECID wdc 753  wal 1257  wnf 1365  wex 1397  [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-dc 754  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662
This theorem is referenced by:  mo2dc  1971
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