Theorem wl-ax11-lem2 32542

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

Assertion

Ref	Expression
wl-ax11-lem2	⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ∀𝑥 𝑢 = 𝑦)

Distinct variable group:   𝑥,𝑢

Proof of Theorem wl-ax11-lem2

Step	Hyp	Ref	Expression
1		sp 2041	. . 3 ⊢ (∀𝑢 𝑢 = 𝑦 → 𝑢 = 𝑦)
2		aev 1970	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑢 → ∀𝑥 𝑥 = 𝑦)
3		pm2.21 119	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑢))
4	2, 3	impbid2 215	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑢 ↔ ∀𝑥 𝑥 = 𝑦))
5	1, 4	anim12i 588	. 2 ⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑢 = 𝑦 ∧ (∀𝑥 𝑥 = 𝑢 ↔ ∀𝑥 𝑥 = 𝑦)))
6		wl-aleq 32501	. 2 ⊢ (∀𝑥 𝑢 = 𝑦 ↔ (𝑢 = 𝑦 ∧ (∀𝑥 𝑥 = 𝑢 ↔ ∀𝑥 𝑥 = 𝑦)))
7	5, 6	sylibr 223	1 ⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ∀𝑥 𝑢 = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473

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

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-lem3  32543