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Theorem eunexOLD 5138
 Description: Obsolete proof of eunex 5137 as of 2-Jan-2023. (Contributed by NM, 24-Oct-2010.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
eunexOLD (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)

Proof of Theorem eunexOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dtru 5118 . . . . 5 ¬ ∀𝑥 𝑥 = 𝑦
2 alim 1773 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → (∀𝑥𝜑 → ∀𝑥 𝑥 = 𝑦))
31, 2mtoi 191 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
43exlimiv 1889 . . 3 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ¬ ∀𝑥𝜑)
54adantl 474 . 2 ((∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)) → ¬ ∀𝑥𝜑)
6 eu3v 2581 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
7 exnal 1789 . 2 (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
85, 6, 73imtr4i 284 1 (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 387  ∀wal 1505  ∃wex 1742  ∃!weu 2579 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-12 2104  ax-nul 5061  ax-pow 5113 This theorem depends on definitions:  df-bi 199  df-an 388  df-tru 1510  df-ex 1743  df-nf 1747  df-mo 2544  df-eu 2580 This theorem is referenced by: (None)
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