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Theorem eunex 4444
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 1491 . . 3 𝑦𝜑
21eu3 2021 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
3 dtruex 4442 . . . . 5 𝑥 ¬ 𝑥 = 𝑦
4 nfa1 1504 . . . . . 6 𝑥𝑥(𝜑𝑥 = 𝑦)
5 sp 1471 . . . . . . 7 (∀𝑥(𝜑𝑥 = 𝑦) → (𝜑𝑥 = 𝑦))
65con3d 603 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → (¬ 𝑥 = 𝑦 → ¬ 𝜑))
74, 6eximd 1574 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (∃𝑥 ¬ 𝑥 = 𝑦 → ∃𝑥 ¬ 𝜑))
83, 7mpi 15 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → ∃𝑥 ¬ 𝜑)
98exlimiv 1560 . . 3 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑥 ¬ 𝜑)
109adantl 273 . 2 ((∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)) → ∃𝑥 ¬ 𝜑)
112, 10sylbi 120 1 (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wal 1312   = wceq 1314  wex 1451  ∃!weu 1975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-v 2660  df-dif 3041  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501
This theorem is referenced by: (None)
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