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Theorem wl-ax8clv1 33508
 Description: Lifting the distinct variable constraint on 𝑥 and 𝑦 in ax-wl-8cl 33507. (Contributed by Wolf Lammen, 27-Nov-2021.)
Assertion
Ref Expression
wl-ax8clv1 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

Proof of Theorem wl-ax8clv1
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 equvinv 2003 . 2 (𝑥 = 𝑦 ↔ ∃𝑢(𝑢 = 𝑥𝑢 = 𝑦))
2 ax-wl-8cl 33507 . . . . 5 (𝑥 = 𝑢 → (𝑥𝐴𝑢𝐴))
32equcoms 1993 . . . 4 (𝑢 = 𝑥 → (𝑥𝐴𝑢𝐴))
4 ax-wl-8cl 33507 . . . 4 (𝑢 = 𝑦 → (𝑢𝐴𝑦𝐴))
53, 4sylan9 690 . . 3 ((𝑢 = 𝑥𝑢 = 𝑦) → (𝑥𝐴𝑦𝐴))
65exlimiv 1898 . 2 (∃𝑢(𝑢 = 𝑥𝑢 = 𝑦) → (𝑥𝐴𝑦𝐴))
71, 6sylbi 207 1 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1744   ∈ wcel-wl 33503 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-wl-8cl 33507 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745 This theorem is referenced by: (None)
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