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Theorem wl-euequf 33895
 Description: euequ 2669 proved with a distinctor. (Contributed by Wolf Lammen, 23-Sep-2020.)
Assertion
Ref Expression
wl-euequf (¬ ∀𝑥 𝑥 = 𝑦 → ∃!𝑥 𝑥 = 𝑦)

Proof of Theorem wl-euequf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ax6ev 2077 . . 3 𝑧 𝑧 = 𝑦
2 nfv 2013 . . . 4 𝑧 ¬ ∀𝑥 𝑥 = 𝑦
3 nfna1 2203 . . . . 5 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
4 nfeqf2 2394 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
5 equequ2 2130 . . . . . . 7 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
65equcoms 2124 . . . . . 6 (𝑧 = 𝑦 → (𝑥 = 𝑦𝑥 = 𝑧))
76a1i 11 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝑥 = 𝑦𝑥 = 𝑧)))
83, 4, 7alrimdd 2257 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑦𝑥 = 𝑧)))
92, 8eximd 2259 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑧 𝑧 = 𝑦 → ∃𝑧𝑥(𝑥 = 𝑦𝑥 = 𝑧)))
101, 9mpi 20 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑧𝑥(𝑥 = 𝑦𝑥 = 𝑧))
11 eu6 2645 . 2 (∃!𝑥 𝑥 = 𝑦 ↔ ∃𝑧𝑥(𝑥 = 𝑦𝑥 = 𝑧))
1210, 11sylibr 226 1 (¬ ∀𝑥 𝑥 = 𝑦 → ∃!𝑥 𝑥 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198  ∀wal 1654  ∃wex 1878  ∃!weu 2639 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-ex 1879  df-nf 1883  df-mo 2605  df-eu 2640 This theorem is referenced by: (None)
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