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Theorem wl-euequ1f 32983
Description: euequ1 2480 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 1892 . . 3 𝑧 𝑧 = 𝑦
2 nfv 1845 . . . 4 𝑧 ¬ ∀𝑥 𝑥 = 𝑦
3 nfnae 2322 . . . . 5 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
4 nfeqf2 2301 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
5 equequ2 1955 . . . . . . 7 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
65equcoms 1949 . . . . . 6 (𝑧 = 𝑦 → (𝑥 = 𝑦𝑥 = 𝑧))
76a1i 11 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → (𝑥 = 𝑦𝑥 = 𝑧)))
83, 4, 7alrimdd 2086 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑦𝑥 = 𝑧)))
92, 8eximd 2088 . . 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 2478 . 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 1478  wex 1701  ∃!weu 2474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-eu 2478
This theorem is referenced by: (None)
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