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Theorem eunex 4377
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 1466 . . 3 𝑦𝜑
21eu3 1994 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
3 dtruex 4375 . . . . 5 𝑥 ¬ 𝑥 = 𝑦
4 nfa1 1479 . . . . . 6 𝑥𝑥(𝜑𝑥 = 𝑦)
5 sp 1446 . . . . . . 7 (∀𝑥(𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
65con3d 596 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → (¬ 𝑥 = 𝑦 → ¬ 𝜑))
74, 6eximd 1548 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (∃𝑥 ¬ 𝑥 = 𝑦 → ∃𝑥 ¬ 𝜑))
83, 7mpi 15 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → ∃𝑥 ¬ 𝜑)
98exlimiv 1534 . . 3 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑥 ¬ 𝜑)
109adantl 271 . 2 ((∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)) → ∃𝑥 ¬ 𝜑)
112, 10sylbi 119 1 (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wal 1287   = wceq 1289  wex 1426  ∃!weu 1948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-setind 4353
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-v 2621  df-dif 3001  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452
This theorem is referenced by: (None)
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