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Theorem wl-ax11-lem3 33035
Description: Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)
Assertion
Ref Expression
wl-ax11-lem3 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑢 𝑢 = 𝑦)
Distinct variable group:   𝑥,𝑢

Proof of Theorem wl-ax11-lem3
StepHypRef Expression
1 nfna1 2026 . 2 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
2 wl-naev 32973 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑢 𝑢 = 𝑥)
3 nfa1 2025 . . . . . . 7 𝑢𝑢 𝑢 = 𝑦
4 nfna1 2026 . . . . . . 7 𝑢 ¬ ∀𝑢 𝑢 = 𝑥
53, 4nfan 1825 . . . . . 6 𝑢(∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑢 𝑢 = 𝑥)
6 axc11n 2306 . . . . . . . . . . 11 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
7 wl-aetr 32988 . . . . . . . . . . 11 (∀𝑦 𝑦 = 𝑢 → (∀𝑦 𝑦 = 𝑥 → ∀𝑢 𝑢 = 𝑥))
86, 7syl5 34 . . . . . . . . . 10 (∀𝑦 𝑦 = 𝑢 → (∀𝑥 𝑥 = 𝑦 → ∀𝑢 𝑢 = 𝑥))
98aecoms 2311 . . . . . . . . 9 (∀𝑢 𝑢 = 𝑦 → (∀𝑥 𝑥 = 𝑦 → ∀𝑢 𝑢 = 𝑥))
109con3d 148 . . . . . . . 8 (∀𝑢 𝑢 = 𝑦 → (¬ ∀𝑢 𝑢 = 𝑥 → ¬ ∀𝑥 𝑥 = 𝑦))
1110imdistani 725 . . . . . . 7 ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑢 𝑢 = 𝑥) → (∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦))
12 wl-ax11-lem2 33034 . . . . . . 7 ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ∀𝑥 𝑢 = 𝑦)
1311, 12syl 17 . . . . . 6 ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑢 𝑢 = 𝑥) → ∀𝑥 𝑢 = 𝑦)
145, 13alrimi 2080 . . . . 5 ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑢 𝑢 = 𝑥) → ∀𝑢𝑥 𝑢 = 𝑦)
152, 14sylan2 491 . . . 4 ((∀𝑢 𝑢 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → ∀𝑢𝑥 𝑢 = 𝑦)
1615expcom 451 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑢 𝑢 = 𝑦 → ∀𝑢𝑥 𝑢 = 𝑦))
17 ax-wl-11v 33032 . . 3 (∀𝑢𝑥 𝑢 = 𝑦 → ∀𝑥𝑢 𝑢 = 𝑦)
1816, 17syl6 35 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑢 𝑢 = 𝑦 → ∀𝑥𝑢 𝑢 = 𝑦))
191, 18nf5d 2115 1 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑢 𝑢 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wal 1478  wnf 1705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044  ax-13 2245  ax-wl-11v 33032
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707
This theorem is referenced by:  wl-ax11-lem4  33036  wl-ax11-lem6  33038
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