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Theorem wl-euequf 34809
Description: euequ 2679 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 1968 . . 3 𝑧 𝑧 = 𝑦
2 nfv 1911 . . . 4 𝑧 ¬ ∀𝑥 𝑥 = 𝑦
3 nfna1 2152 . . . . 5 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
4 nfeqf2 2391 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
5 equequ2 2029 . . . . . . 7 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
65equcoms 2023 . . . . . 6 (𝑧 = 𝑦 → (𝑥 = 𝑦𝑥 = 𝑧))
76a1i 11 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝑥 = 𝑦𝑥 = 𝑧)))
83, 4, 7alrimdd 2210 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑦𝑥 = 𝑧)))
92, 8eximd 2212 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑧 𝑧 = 𝑦 → ∃𝑧𝑥(𝑥 = 𝑦𝑥 = 𝑧)))
101, 9mpi 20 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑧𝑥(𝑥 = 𝑦𝑥 = 𝑧))
11 eu6 2655 . 2 (∃!𝑥 𝑥 = 𝑦 ↔ ∃𝑧𝑥(𝑥 = 𝑦𝑥 = 𝑧))
1210, 11sylibr 236 1 (¬ ∀𝑥 𝑥 = 𝑦 → ∃!𝑥 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wal 1531  wex 1776  ∃!weu 2649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-12 2173  ax-13 2386
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-nf 1781  df-mo 2618  df-eu 2650
This theorem is referenced by: (None)
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