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Theorem wl-ax13lem1 33619
Description: A version of ax-wl-13v 33618 with one distinct variable restriction dropped. For convenience, 𝑦 is kept on the right side of equations. This proof bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 23-Jul-2021.)
Assertion
Ref Expression
wl-ax13lem1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Distinct variable group:   𝑥,𝑧

Proof of Theorem wl-ax13lem1
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 equvinva 2116 . 2 (𝑧 = 𝑦 → ∃𝑤(𝑧 = 𝑤𝑦 = 𝑤))
2 ax-wl-13v 33618 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑤 → ∀𝑥 𝑦 = 𝑤))
3 equeucl 2108 . . . . . 6 (𝑧 = 𝑤 → (𝑦 = 𝑤𝑧 = 𝑦))
43alimdv 1996 . . . . 5 (𝑧 = 𝑤 → (∀𝑥 𝑦 = 𝑤 → ∀𝑥 𝑧 = 𝑦))
52, 4syl9 77 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑤 → (𝑦 = 𝑤 → ∀𝑥 𝑧 = 𝑦)))
65impd 396 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ((𝑧 = 𝑤𝑦 = 𝑤) → ∀𝑥 𝑧 = 𝑦))
76exlimdv 2012 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑤(𝑧 = 𝑤𝑦 = 𝑤) → ∀𝑥 𝑧 = 𝑦))
81, 7syl5 34 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wal 1628  wex 1851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-wl-13v 33618
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1852
This theorem is referenced by: (None)
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