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Theorem mo2n 1944
Description: There is at most one of something which does not exist. (Contributed by Jim Kingdon, 2-Jul-2018.)
Hypothesis
Ref Expression
mon.1 𝑦𝜑
Assertion
Ref Expression
mo2n (¬ ∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem mo2n
StepHypRef Expression
1 mon.1 . . 3 𝑦𝜑
21sb8e 1753 . 2 (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑)
3 alnex 1404 . . 3 (∀𝑦 ¬ [𝑦 / 𝑥]𝜑 ↔ ¬ ∃𝑦[𝑦 / 𝑥]𝜑)
4 nfs1v 1831 . . . . . 6 𝑥[𝑦 / 𝑥]𝜑
54nfn 1564 . . . . 5 𝑥 ¬ [𝑦 / 𝑥]𝜑
61nfn 1564 . . . . 5 𝑦 ¬ 𝜑
7 sbequ1 1667 . . . . . . 7 (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))
87equcoms 1610 . . . . . 6 (𝑦 = 𝑥 → (𝜑 → [𝑦 / 𝑥]𝜑))
98con3d 571 . . . . 5 (𝑦 = 𝑥 → (¬ [𝑦 / 𝑥]𝜑 → ¬ 𝜑))
105, 6, 9cbv3 1646 . . . 4 (∀𝑦 ¬ [𝑦 / 𝑥]𝜑 → ∀𝑥 ¬ 𝜑)
11 pm2.21 557 . . . . 5 𝜑 → (𝜑𝑥 = 𝑦))
1211alimi 1360 . . . 4 (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑𝑥 = 𝑦))
13 19.8a 1498 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
1410, 12, 133syl 17 . . 3 (∀𝑦 ¬ [𝑦 / 𝑥]𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
153, 14sylbir 129 . 2 (¬ ∃𝑦[𝑦 / 𝑥]𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
162, 15sylnbi 613 1 (¬ ∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  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-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
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662
This theorem is referenced by:  modc  1959
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