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Theorem kmlem6 9015
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 3093 . 2 (∀𝑧𝑥 (𝑧 ≠ ∅ ∧ ∀𝑤𝑥 (𝜑𝐴 = ∅)) ↔ (∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝜑𝐴 = ∅)))
2 n0 3964 . . . . 5 (𝑧 ≠ ∅ ↔ ∃𝑣 𝑣𝑧)
32biimpi 206 . . . 4 (𝑧 ≠ ∅ → ∃𝑣 𝑣𝑧)
4 ne0i 3954 . . . . . . . 8 (𝑣𝐴𝐴 ≠ ∅)
54necon2bi 2853 . . . . . . 7 (𝐴 = ∅ → ¬ 𝑣𝐴)
65imim2i 16 . . . . . 6 ((𝜑𝐴 = ∅) → (𝜑 → ¬ 𝑣𝐴))
76ralimi 2981 . . . . 5 (∀𝑤𝑥 (𝜑𝐴 = ∅) → ∀𝑤𝑥 (𝜑 → ¬ 𝑣𝐴))
87alrimiv 1895 . . . 4 (∀𝑤𝑥 (𝜑𝐴 = ∅) → ∀𝑣𝑤𝑥 (𝜑 → ¬ 𝑣𝐴))
9 19.29r 1842 . . . . 5 ((∃𝑣 𝑣𝑧 ∧ ∀𝑣𝑤𝑥 (𝜑 → ¬ 𝑣𝐴)) → ∃𝑣(𝑣𝑧 ∧ ∀𝑤𝑥 (𝜑 → ¬ 𝑣𝐴)))
10 df-rex 2947 . . . . 5 (∃𝑣𝑧𝑤𝑥 (𝜑 → ¬ 𝑣𝐴) ↔ ∃𝑣(𝑣𝑧 ∧ ∀𝑤𝑥 (𝜑 → ¬ 𝑣𝐴)))
119, 10sylibr 224 . . . 4 ((∃𝑣 𝑣𝑧 ∧ ∀𝑣𝑤𝑥 (𝜑 → ¬ 𝑣𝐴)) → ∃𝑣𝑧𝑤𝑥 (𝜑 → ¬ 𝑣𝐴))
123, 8, 11syl2an 493 . . 3 ((𝑧 ≠ ∅ ∧ ∀𝑤𝑥 (𝜑𝐴 = ∅)) → ∃𝑣𝑧𝑤𝑥 (𝜑 → ¬ 𝑣𝐴))
1312ralimi 2981 . 2 (∀𝑧𝑥 (𝑧 ≠ ∅ ∧ ∀𝑤𝑥 (𝜑𝐴 = ∅)) → ∀𝑧𝑥𝑣𝑧𝑤𝑥 (𝜑 → ¬ 𝑣𝐴))
141, 13sylbir 225 1 ((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝜑𝐴 = ∅)) → ∀𝑧𝑥𝑣𝑧𝑤𝑥 (𝜑 → ¬ 𝑣𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wal 1521   = wceq 1523  wex 1744  wcel 2030  wne 2823  wral 2941  wrex 2942  c0 3948
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-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-nul 3949
This theorem is referenced by:  kmlem7  9016
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