Theorem wl-ax11-lem4 33697

Description: Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)

Assertion

Ref	Expression
wl-ax11-lem4	⊢ Ⅎ𝑥(∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦)

Distinct variable group:   𝑥,𝑢

Proof of Theorem wl-ax11-lem4

Step	Hyp	Ref	Expression
1		ancom 465	. 2 ⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) ↔ (¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑢 𝑢 = 𝑦))
2		nfna1 2179	. . 3 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
3		wl-ax11-lem3 33696	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑢 𝑢 = 𝑦)
4	2, 3	nfan1 2216	. 2 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑢 𝑢 = 𝑦)
5	1, 4	nfxfr 1928	1 ⊢ Ⅎ𝑥(∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦)

Syntax hints:  ¬ wn 3   ∧ wa 383  ∀wal 1630  Ⅎwnf 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