Theorem wl-ax11-lem3 37182

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

Assertion

Ref	Expression
wl-ax11-lem3	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑢 𝑢 = 𝑦)

Distinct variable group:   𝑥,𝑢

Proof of Theorem wl-ax11-lem3

Step	Hyp	Ref	Expression
1		nfna1 2141	. 2 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
2		naev 2055	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑢 𝑢 = 𝑥)
3		nfa1 2140	. . . . . . 7 ⊢ Ⅎ𝑢∀𝑢 𝑢 = 𝑦
4		nfna1 2141	. . . . . . 7 ⊢ Ⅎ𝑢 ¬ ∀𝑢 𝑢 = 𝑥
5	3, 4	nfan 1894	. . . . . 6 ⊢ Ⅎ𝑢(∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑢 𝑢 = 𝑥)
6		axc11n 2419	. . . . . . . . . . 11 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
7		wl-aetr 37124	. . . . . . . . . . 11 ⊢ (∀𝑦 𝑦 = 𝑢 → (∀𝑦 𝑦 = 𝑥 → ∀𝑢 𝑢 = 𝑥))
8	6, 7	syl5 34	. . . . . . . . . 10 ⊢ (∀𝑦 𝑦 = 𝑢 → (∀𝑥 𝑥 = 𝑦 → ∀𝑢 𝑢 = 𝑥))
9	8	aecoms 2421	. . . . . . . . 9 ⊢ (∀𝑢 𝑢 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑢 𝑢 = 𝑥))
10	9	con3d 152	. . . . . . . 8 ⊢ (∀𝑢 𝑢 = 𝑦 → (¬ ∀𝑢 𝑢 = 𝑥 → ¬ ∀𝑥 𝑥 = 𝑦))
11	10	imdistani 567	. . . . . . 7 ⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑢 𝑢 = 𝑥) → (∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦))
12		wl-ax11-lem2 37181	. . . . . . 7 ⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ∀𝑥 𝑢 = 𝑦)
13	11, 12	syl 17	. . . . . 6 ⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑢 𝑢 = 𝑥) → ∀𝑥 𝑢 = 𝑦)
14	5, 13	alrimi 2201	. . . . 5 ⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑢 𝑢 = 𝑥) → ∀𝑢∀𝑥 𝑢 = 𝑦)
15	2, 14	sylan2 591	. . . 4 ⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ∀𝑢∀𝑥 𝑢 = 𝑦)
16	15	expcom 412	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑢 𝑢 = 𝑦 → ∀𝑢∀𝑥 𝑢 = 𝑦))
17		ax-wl-11v 37179	. . 3 ⊢ (∀𝑢∀𝑥 𝑢 = 𝑦 → ∀𝑥∀𝑢 𝑢 = 𝑦)
18	16, 17	syl6 35	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑢 𝑢 = 𝑦 → ∀𝑥∀𝑢 𝑢 = 𝑦))
19	1, 18	nf5d 2273	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑢 𝑢 = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394  ∀wal 1531  Ⅎwnf 1777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2166  ax-13 2365  ax-wl-11v 37179

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778

This theorem is referenced by:  wl-ax11-lem4  37183  wl-ax11-lem6  37185