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Theorem eunex 4578
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 1539 . . 3 𝑦𝜑
21eu3 2084 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
3 dtruex 4576 . . . . 5 𝑥 ¬ 𝑥 = 𝑦
4 nfa1 1552 . . . . . 6 𝑥𝑥(𝜑𝑥 = 𝑦)
5 sp 1522 . . . . . . 7 (∀𝑥(𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
65con3d 632 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → (¬ 𝑥 = 𝑦 → ¬ 𝜑))
74, 6eximd 1623 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (∃𝑥 ¬ 𝑥 = 𝑦 → ∃𝑥 ¬ 𝜑))
83, 7mpi 15 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → ∃𝑥 ¬ 𝜑)
98exlimiv 1609 . . 3 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑥 ¬ 𝜑)
109adantl 277 . 2 ((∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)) → ∃𝑥 ¬ 𝜑)
112, 10sylbi 121 1 (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wal 1362   = wceq 1364  wex 1503  ∃!weu 2038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613
This theorem is referenced by: (None)
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