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Theorem eunex 4285
 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 1421 . . 3 𝑦𝜑
21eu3 1946 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
3 dtruex 4283 . . . . 5 𝑥 ¬ 𝑥 = 𝑦
4 nfa1 1434 . . . . . 6 𝑥𝑥(𝜑𝑥 = 𝑦)
5 sp 1401 . . . . . . 7 (∀𝑥(𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
65con3d 561 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → (¬ 𝑥 = 𝑦 → ¬ 𝜑))
74, 6eximd 1503 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (∃𝑥 ¬ 𝑥 = 𝑦 → ∃𝑥 ¬ 𝜑))
83, 7mpi 15 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → ∃𝑥 ¬ 𝜑)
98exlimiv 1489 . . 3 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑥 ¬ 𝜑)
109adantl 262 . 2 ((∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)) → ∃𝑥 ¬ 𝜑)
112, 10sylbi 114 1 (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97  ∀wal 1241   = wceq 1243  ∃wex 1381  ∃!weu 1900 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-setind 4262 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381 This theorem is referenced by: (None)
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