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Theorem wl-ax11-lem4 32544
 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 2016 . . 3 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
3 wl-ax11-lem3 32543 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑢 𝑢 = 𝑦)
42, 3nfan1 2056 . 2 𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑢 𝑢 = 𝑦)
51, 4nfxfr 1771 1 𝑥(∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 383  ∀wal 1473  Ⅎwnf 1699 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234  ax-wl-11v 32540 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701 This theorem is referenced by:  wl-ax11-lem8  32548
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