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Theorem bnj1176 30834
 Description: Technical lemma for bnj69 30839. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1176.51 ((𝜑𝜓) → (𝑅 Fr 𝐴𝐶𝐴𝐶 ≠ ∅ ∧ 𝐶 ∈ V))
bnj1176.52 ((𝑅 Fr 𝐴𝐶𝐴𝐶 ≠ ∅ ∧ 𝐶 ∈ V) → ∃𝑧𝐶𝑤𝐶 ¬ 𝑤𝑅𝑧)
Assertion
Ref Expression
bnj1176 𝑧𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶))))
Distinct variable groups:   𝑤,𝐶   𝜑,𝑤,𝑧   𝜓,𝑤,𝑧
Allowed substitution hints:   𝜃(𝑧,𝑤)   𝐴(𝑧,𝑤)   𝐶(𝑧)   𝑅(𝑧,𝑤)

Proof of Theorem bnj1176
StepHypRef Expression
1 bnj1176.51 . . . . . . . . 9 ((𝜑𝜓) → (𝑅 Fr 𝐴𝐶𝐴𝐶 ≠ ∅ ∧ 𝐶 ∈ V))
2 bnj1176.52 . . . . . . . . 9 ((𝑅 Fr 𝐴𝐶𝐴𝐶 ≠ ∅ ∧ 𝐶 ∈ V) → ∃𝑧𝐶𝑤𝐶 ¬ 𝑤𝑅𝑧)
31, 2syl 17 . . . . . . . 8 ((𝜑𝜓) → ∃𝑧𝐶𝑤𝐶 ¬ 𝑤𝑅𝑧)
4 df-ral 2913 . . . . . . . . 9 (∀𝑤𝐶 ¬ 𝑤𝑅𝑧 ↔ ∀𝑤(𝑤𝐶 → ¬ 𝑤𝑅𝑧))
54rexbii 3036 . . . . . . . 8 (∃𝑧𝐶𝑤𝐶 ¬ 𝑤𝑅𝑧 ↔ ∃𝑧𝐶𝑤(𝑤𝐶 → ¬ 𝑤𝑅𝑧))
63, 5sylib 208 . . . . . . 7 ((𝜑𝜓) → ∃𝑧𝐶𝑤(𝑤𝐶 → ¬ 𝑤𝑅𝑧))
7 df-rex 2914 . . . . . . 7 (∃𝑧𝐶𝑤(𝑤𝐶 → ¬ 𝑤𝑅𝑧) ↔ ∃𝑧(𝑧𝐶 ∧ ∀𝑤(𝑤𝐶 → ¬ 𝑤𝑅𝑧)))
86, 7sylib 208 . . . . . 6 ((𝜑𝜓) → ∃𝑧(𝑧𝐶 ∧ ∀𝑤(𝑤𝐶 → ¬ 𝑤𝑅𝑧)))
9 19.28v 1906 . . . . . . 7 (∀𝑤(𝑧𝐶 ∧ (𝑤𝐶 → ¬ 𝑤𝑅𝑧)) ↔ (𝑧𝐶 ∧ ∀𝑤(𝑤𝐶 → ¬ 𝑤𝑅𝑧)))
109exbii 1771 . . . . . 6 (∃𝑧𝑤(𝑧𝐶 ∧ (𝑤𝐶 → ¬ 𝑤𝑅𝑧)) ↔ ∃𝑧(𝑧𝐶 ∧ ∀𝑤(𝑤𝐶 → ¬ 𝑤𝑅𝑧)))
118, 10sylibr 224 . . . . 5 ((𝜑𝜓) → ∃𝑧𝑤(𝑧𝐶 ∧ (𝑤𝐶 → ¬ 𝑤𝑅𝑧)))
12 19.37v 1907 . . . . 5 (∃𝑧((𝜑𝜓) → ∀𝑤(𝑧𝐶 ∧ (𝑤𝐶 → ¬ 𝑤𝑅𝑧))) ↔ ((𝜑𝜓) → ∃𝑧𝑤(𝑧𝐶 ∧ (𝑤𝐶 → ¬ 𝑤𝑅𝑧))))
1311, 12mpbir 221 . . . 4 𝑧((𝜑𝜓) → ∀𝑤(𝑧𝐶 ∧ (𝑤𝐶 → ¬ 𝑤𝑅𝑧)))
14 19.21v 1865 . . . . 5 (∀𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝑤𝐶 → ¬ 𝑤𝑅𝑧))) ↔ ((𝜑𝜓) → ∀𝑤(𝑧𝐶 ∧ (𝑤𝐶 → ¬ 𝑤𝑅𝑧))))
1514exbii 1771 . . . 4 (∃𝑧𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝑤𝐶 → ¬ 𝑤𝑅𝑧))) ↔ ∃𝑧((𝜑𝜓) → ∀𝑤(𝑧𝐶 ∧ (𝑤𝐶 → ¬ 𝑤𝑅𝑧))))
1613, 15mpbir 221 . . 3 𝑧𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝑤𝐶 → ¬ 𝑤𝑅𝑧)))
17 con2b 349 . . . . . . 7 ((𝑤𝐶 → ¬ 𝑤𝑅𝑧) ↔ (𝑤𝑅𝑧 → ¬ 𝑤𝐶))
1817anbi2i 729 . . . . . 6 ((𝑧𝐶 ∧ (𝑤𝐶 → ¬ 𝑤𝑅𝑧)) ↔ (𝑧𝐶 ∧ (𝑤𝑅𝑧 → ¬ 𝑤𝐶)))
1918imbi2i 326 . . . . 5 (((𝜑𝜓) → (𝑧𝐶 ∧ (𝑤𝐶 → ¬ 𝑤𝑅𝑧))) ↔ ((𝜑𝜓) → (𝑧𝐶 ∧ (𝑤𝑅𝑧 → ¬ 𝑤𝐶))))
2019albii 1744 . . . 4 (∀𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝑤𝐶 → ¬ 𝑤𝑅𝑧))) ↔ ∀𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝑤𝑅𝑧 → ¬ 𝑤𝐶))))
2120exbii 1771 . . 3 (∃𝑧𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝑤𝐶 → ¬ 𝑤𝑅𝑧))) ↔ ∃𝑧𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝑤𝑅𝑧 → ¬ 𝑤𝐶))))
2216, 21mpbi 220 . 2 𝑧𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝑤𝑅𝑧 → ¬ 𝑤𝐶)))
23 ax-1 6 . . . . 5 ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)))
2423anim2i 592 . . . 4 ((𝑧𝐶 ∧ (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶))))
2524imim2i 16 . . 3 (((𝜑𝜓) → (𝑧𝐶 ∧ (𝑤𝑅𝑧 → ¬ 𝑤𝐶))) → ((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)))))
2625alimi 1736 . 2 (∀𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝑤𝑅𝑧 → ¬ 𝑤𝐶))) → ∀𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)))))
2722, 26bnj101 30550 1 𝑧𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384  ∀wal 1478  ∃wex 1701   ∈ wcel 1987   ≠ wne 2790  ∀wral 2908  ∃wrex 2909  Vcvv 3190   ⊆ wss 3560  ∅c0 3897   class class class wbr 4623   Fr wfr 5040   ∧ w-bnj17 30512 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 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-ral 2913  df-rex 2914 This theorem is referenced by:  bnj1190  30837
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