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Theorem wl-ax11-lem2 33493
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
StepHypRef Expression
1 sp 2091 . . 3 (∀𝑢 𝑢 = 𝑦𝑢 = 𝑦)
2 aev 2025 . . . 4 (∀𝑥 𝑥 = 𝑢 → ∀𝑥 𝑥 = 𝑦)
3 pm2.21 120 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑢))
42, 3impbid2 216 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑢 ↔ ∀𝑥 𝑥 = 𝑦))
51, 4anim12i 589 . 2 ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑢 = 𝑦 ∧ (∀𝑥 𝑥 = 𝑢 ↔ ∀𝑥 𝑥 = 𝑦)))
6 wl-aleq 33452 . 2 (∀𝑥 𝑢 = 𝑦 ↔ (𝑢 = 𝑦 ∧ (∀𝑥 𝑥 = 𝑢 ↔ ∀𝑥 𝑥 = 𝑦)))
75, 6sylibr 224 1 ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ∀𝑥 𝑢 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750
This theorem is referenced by:  wl-ax11-lem3  33494
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