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Theorem bnj1174 31522
Description: Technical lemma for bnj69 31529. 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
bnj1174.3 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)
bnj1174.59 𝑧𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶))))
bnj1174.74 (𝜃 → (𝑤𝑅𝑧𝑤 ∈ trCl(𝑋, 𝐴, 𝑅)))
Assertion
Ref Expression
bnj1174 𝑧𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))

Proof of Theorem bnj1174
StepHypRef Expression
1 bnj1174.59 . . . . 5 𝑧𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶))))
2 bnj1174.74 . . . . . . . . . . . 12 (𝜃 → (𝑤𝑅𝑧𝑤 ∈ trCl(𝑋, 𝐴, 𝑅)))
3 bnj1174.3 . . . . . . . . . . . . . . . 16 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)
43eleq2i 2836 . . . . . . . . . . . . . . 15 (𝑤𝐶𝑤 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵))
54notbii 311 . . . . . . . . . . . . . 14 𝑤𝐶 ↔ ¬ 𝑤 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵))
6 ianor 1004 . . . . . . . . . . . . . . . . 17 (¬ (𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑤𝐵) ↔ (¬ 𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) ∨ ¬ 𝑤𝐵))
7 elin 3960 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ↔ (𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑤𝐵))
87notbii 311 . . . . . . . . . . . . . . . . 17 𝑤 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ↔ ¬ (𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑤𝐵))
9 pm4.62 882 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) → ¬ 𝑤𝐵) ↔ (¬ 𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) ∨ ¬ 𝑤𝐵))
106, 8, 93bitr4i 294 . . . . . . . . . . . . . . . 16 𝑤 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ↔ (𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) → ¬ 𝑤𝐵))
1110biimpi 207 . . . . . . . . . . . . . . 15 𝑤 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) → (𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) → ¬ 𝑤𝐵))
1211impcom 396 . . . . . . . . . . . . . 14 ((𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ ¬ 𝑤 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)) → ¬ 𝑤𝐵)
135, 12sylan2b 587 . . . . . . . . . . . . 13 ((𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ ¬ 𝑤𝐶) → ¬ 𝑤𝐵)
1413ex 401 . . . . . . . . . . . 12 (𝑤 ∈ trCl(𝑋, 𝐴, 𝑅) → (¬ 𝑤𝐶 → ¬ 𝑤𝐵))
152, 14syl6 35 . . . . . . . . . . 11 (𝜃 → (𝑤𝑅𝑧 → (¬ 𝑤𝐶 → ¬ 𝑤𝐵)))
1615a2d 29 . . . . . . . . . 10 (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))
1716biantru 525 . . . . . . . . 9 ((𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶))) ↔ ((𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶))) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
18 df-3an 1109 . . . . . . . . 9 ((𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) ↔ ((𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶))) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
19 3anass 1116 . . . . . . . . 9 ((𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) ↔ (𝑧𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))))
2017, 18, 193bitr2i 290 . . . . . . . 8 ((𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶))) ↔ (𝑧𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))))
2120imbi2i 327 . . . . . . 7 (((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)))) ↔ ((𝜑𝜓) → (𝑧𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))))
2221albii 1914 . . . . . 6 (∀𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)))) ↔ ∀𝑤((𝜑𝜓) → (𝑧𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))))
2322exbii 1943 . . . . 5 (∃𝑧𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)))) ↔ ∃𝑧𝑤((𝜑𝜓) → (𝑧𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))))
241, 23mpbi 221 . . . 4 𝑧𝑤((𝜑𝜓) → (𝑧𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))))
25 imdi 379 . . . . . . . 8 ((𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))) ↔ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) → (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
26 pm3.35 837 . . . . . . . 8 (((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) → (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) → (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))
2725, 26sylan2b 587 . . . . . . 7 (((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) → (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))
2827anim2i 610 . . . . . 6 ((𝑧𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
2928imim2i 16 . . . . 5 (((𝜑𝜓) → (𝑧𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))) → ((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
3029alimi 1906 . . . 4 (∀𝑤((𝜑𝜓) → (𝑧𝐶 ∧ ((𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐶)) ∧ (𝜃 → ((𝑤𝑅𝑧 → ¬ 𝑤𝐶) → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))) → ∀𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
3124, 30bnj101 31243 . . 3 𝑧𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
32 ancl 540 . . . . 5 (((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) → ((𝜑𝜓) → ((𝜑𝜓) ∧ (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))))
33 bnj256 31226 . . . . 5 ((𝜑𝜓𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))) ↔ ((𝜑𝜓) ∧ (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
3432, 33syl6ibr 243 . . . 4 (((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) → ((𝜑𝜓) → (𝜑𝜓𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
3534alimi 1906 . . 3 (∀𝑤((𝜑𝜓) → (𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) → ∀𝑤((𝜑𝜓) → (𝜑𝜓𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
3631, 35bnj101 31243 . 2 𝑧𝑤((𝜑𝜓) → (𝜑𝜓𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
37 df-bnj17 31207 . . . . 5 ((𝜑𝜓𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))) ↔ ((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
3837imbi2i 327 . . . 4 (((𝜑𝜓) → (𝜑𝜓𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) ↔ ((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
3938albii 1914 . . 3 (∀𝑤((𝜑𝜓) → (𝜑𝜓𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) ↔ ∀𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
4039exbii 1943 . 2 (∃𝑧𝑤((𝜑𝜓) → (𝜑𝜓𝑧𝐶 ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))) ↔ ∃𝑧𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵)))))
4136, 40mpbi 221 1 𝑧𝑤((𝜑𝜓) → ((𝜑𝜓𝑧𝐶) ∧ (𝜃 → (𝑤𝑅𝑧 → ¬ 𝑤𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wo 873  w3a 1107  wal 1650   = wceq 1652  wex 1874  wcel 2155  cin 3733   class class class wbr 4811  w-bnj17 31206   trClc-bnj18 31214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-in 3741  df-bnj17 31207
This theorem is referenced by:  bnj1190  31527
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