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Theorem eunex 4313
Description: Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by Jim Kingdon, 29-Dec-2018.)
Assertion
Ref Expression
eunex (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)

Proof of Theorem eunex
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfv 1437 . . 3 𝑦𝜑
21eu3 1962 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
3 dtruex 4311 . . . . 5 𝑥 ¬ 𝑥 = 𝑦
4 nfa1 1450 . . . . . 6 𝑥𝑥(𝜑𝑥 = 𝑦)
5 sp 1417 . . . . . . 7 (∀𝑥(𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
65con3d 571 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → (¬ 𝑥 = 𝑦 → ¬ 𝜑))
74, 6eximd 1519 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (∃𝑥 ¬ 𝑥 = 𝑦 → ∃𝑥 ¬ 𝜑))
83, 7mpi 15 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → ∃𝑥 ¬ 𝜑)
98exlimiv 1505 . . 3 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑥 ¬ 𝜑)
109adantl 266 . 2 ((∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)) → ∃𝑥 ¬ 𝜑)
112, 10sylbi 118 1 (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wal 1257   = wceq 1259  wex 1397  ∃!weu 1916
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-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-setind 4290
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-v 2576  df-dif 2948  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409
This theorem is referenced by: (None)
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