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Theorem wl-clelv2-just 33868
Description: Show that the definition df-wl-clelv2 33869 is conservative. (Contributed by Wolf Lammen, 27-Nov-2021.)
Assertion
Ref Expression
wl-clelv2-just (𝑥𝐴 ↔ ∀𝑢(𝑢 = 𝑥𝑢𝐴))
Distinct variable group:   𝑥,𝑢,𝐴

Proof of Theorem wl-clelv2-just
StepHypRef Expression
1 ax-wl-8cl 33866 . . . 4 (𝑢 = 𝑥 → (𝑢𝐴𝑥𝐴))
2 ax-wl-8cl 33866 . . . . 5 (𝑥 = 𝑢 → (𝑥𝐴𝑢𝐴))
32equcoms 2119 . . . 4 (𝑢 = 𝑥 → (𝑥𝐴𝑢𝐴))
41, 3impbid 204 . . 3 (𝑢 = 𝑥 → (𝑢𝐴𝑥𝐴))
54equsalvw 2103 . 2 (∀𝑢(𝑢 = 𝑥𝑢𝐴) ↔ 𝑥𝐴)
65bicomi 216 1 (𝑥𝐴 ↔ ∀𝑢(𝑢 = 𝑥𝑢𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1651  wcel-wl 33862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-wl-8cl 33866
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876
This theorem is referenced by: (None)
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