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Theorem kmlem6 9616
 Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 4 => 1. (Contributed by NM, 26-Mar-2004.)
Assertion
Ref Expression
kmlem6 ((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝜑𝐴 = ∅)) → ∀𝑧𝑥𝑣𝑧𝑤𝑥 (𝜑 → ¬ 𝑣𝐴))
Distinct variable groups:   𝑣,𝐴   𝑥,𝑣,𝜑   𝑤,𝑣,𝑧,𝑥
Allowed substitution hints:   𝜑(𝑧,𝑤)   𝐴(𝑥,𝑧,𝑤)

Proof of Theorem kmlem6
StepHypRef Expression
1 r19.26 3102 . 2 (∀𝑧𝑥 (𝑧 ≠ ∅ ∧ ∀𝑤𝑥 (𝜑𝐴 = ∅)) ↔ (∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝜑𝐴 = ∅)))
2 n0 4246 . . . . 5 (𝑧 ≠ ∅ ↔ ∃𝑣 𝑣𝑧)
32biimpi 219 . . . 4 (𝑧 ≠ ∅ → ∃𝑣 𝑣𝑧)
4 ne0i 4234 . . . . . . . 8 (𝑣𝐴𝐴 ≠ ∅)
54necon2bi 2982 . . . . . . 7 (𝐴 = ∅ → ¬ 𝑣𝐴)
65imim2i 16 . . . . . 6 ((𝜑𝐴 = ∅) → (𝜑 → ¬ 𝑣𝐴))
76ralimi 3093 . . . . 5 (∀𝑤𝑥 (𝜑𝐴 = ∅) → ∀𝑤𝑥 (𝜑 → ¬ 𝑣𝐴))
87alrimiv 1929 . . . 4 (∀𝑤𝑥 (𝜑𝐴 = ∅) → ∀𝑣𝑤𝑥 (𝜑 → ¬ 𝑣𝐴))
9 19.29r 1876 . . . . 5 ((∃𝑣 𝑣𝑧 ∧ ∀𝑣𝑤𝑥 (𝜑 → ¬ 𝑣𝐴)) → ∃𝑣(𝑣𝑧 ∧ ∀𝑤𝑥 (𝜑 → ¬ 𝑣𝐴)))
10 df-rex 3077 . . . . 5 (∃𝑣𝑧𝑤𝑥 (𝜑 → ¬ 𝑣𝐴) ↔ ∃𝑣(𝑣𝑧 ∧ ∀𝑤𝑥 (𝜑 → ¬ 𝑣𝐴)))
119, 10sylibr 237 . . . 4 ((∃𝑣 𝑣𝑧 ∧ ∀𝑣𝑤𝑥 (𝜑 → ¬ 𝑣𝐴)) → ∃𝑣𝑧𝑤𝑥 (𝜑 → ¬ 𝑣𝐴))
123, 8, 11syl2an 599 . . 3 ((𝑧 ≠ ∅ ∧ ∀𝑤𝑥 (𝜑𝐴 = ∅)) → ∃𝑣𝑧𝑤𝑥 (𝜑 → ¬ 𝑣𝐴))
1312ralimi 3093 . 2 (∀𝑧𝑥 (𝑧 ≠ ∅ ∧ ∀𝑤𝑥 (𝜑𝐴 = ∅)) → ∀𝑧𝑥𝑣𝑧𝑤𝑥 (𝜑 → ¬ 𝑣𝐴))
141, 13sylbir 238 1 ((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝜑𝐴 = ∅)) → ∀𝑧𝑥𝑣𝑧𝑤𝑥 (𝜑 → ¬ 𝑣𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2112   ≠ wne 2952  ∀wral 3071  ∃wrex 3072  ∅c0 4226 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ne 2953  df-ral 3076  df-rex 3077  df-dif 3862  df-nul 4227 This theorem is referenced by:  kmlem7  9617
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