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Theorem wl-ax11-lem4 33697
Description: Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
Assertion
Ref Expression
wl-ax11-lem4 𝑥(∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
Distinct variable group:   𝑥,𝑢

Proof of Theorem wl-ax11-lem4
StepHypRef Expression
1 ancom 465 . 2 ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) ↔ (¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑢 𝑢 = 𝑦))
2 nfna1 2179 . . 3 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
3 wl-ax11-lem3 33696 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑢 𝑢 = 𝑦)
42, 3nfan1 2216 . 2 𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑢 𝑢 = 𝑦)
51, 4nfxfr 1928 1 𝑥(∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383  wal 1630  wnf 1857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-10 2169  ax-12 2197  ax-13 2392  ax-wl-11v 33693
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859
This theorem is referenced by:  wl-ax11-lem8  33701
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