Theorem wl-ax11-lem2 33690

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 2206	. . 3 ⊢ (∀𝑢 𝑢 = 𝑦 → 𝑢 = 𝑦)
2		aev 2139	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑢 → ∀𝑥 𝑥 = 𝑦)
3		pm2.21 121	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑢))
4	2, 3	impbid2 216	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑢 ↔ ∀𝑥 𝑥 = 𝑦))
5	1, 4	anim12i 592	. 2 ⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑢 = 𝑦 ∧ (∀𝑥 𝑥 = 𝑢 ↔ ∀𝑥 𝑥 = 𝑦)))
6		wl-aleq 33650	. 2 ⊢ (∀𝑥 𝑢 = 𝑦 ↔ (𝑢 = 𝑦 ∧ (∀𝑥 𝑥 = 𝑢 ↔ ∀𝑥 𝑥 = 𝑦)))
7	5, 6	sylibr 224	1 ⊢ ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ∀𝑥 𝑢 = 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382  ∀wal 1628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-10 2173  ax-12 2202  ax-13 2407

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857

This theorem is referenced by:  wl-ax11-lem3  33691