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Theorem wl-euequ1f 33486
Description: euequ1 2504 proved with a distinctor. (Contributed by Wolf Lammen, 23-Sep-2020.)
Assertion
Ref Expression
wl-euequ1f (¬ ∀𝑥 𝑥 = 𝑦 → ∃!𝑥 𝑥 = 𝑦)

Proof of Theorem wl-euequ1f
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ax6ev 1947 . . 3 𝑧 𝑧 = 𝑦
2 nfv 1883 . . . 4 𝑧 ¬ ∀𝑥 𝑥 = 𝑦
3 nfnae 2351 . . . . 5 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
4 nfeqf2 2333 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
5 equequ2 1999 . . . . . . 7 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
65equcoms 1993 . . . . . 6 (𝑧 = 𝑦 → (𝑥 = 𝑦𝑥 = 𝑧))
76a1i 11 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝑥 = 𝑦𝑥 = 𝑧)))
83, 4, 7alrimdd 2121 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑦𝑥 = 𝑧)))
92, 8eximd 2123 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑧 𝑧 = 𝑦 → ∃𝑧𝑥(𝑥 = 𝑦𝑥 = 𝑧)))
101, 9mpi 20 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑧𝑥(𝑥 = 𝑦𝑥 = 𝑧))
11 df-eu 2502 . 2 (∃!𝑥 𝑥 = 𝑦 ↔ ∃𝑧𝑥(𝑥 = 𝑦𝑥 = 𝑧))
1210, 11sylibr 224 1 (¬ ∀𝑥 𝑥 = 𝑦 → ∃!𝑥 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1521  wex 1744  ∃!weu 2498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-eu 2502
This theorem is referenced by: (None)
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