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Theorem bnj1173 32382
Description: Technical lemma for bnj69 32390. 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
bnj1173.3 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)
bnj1173.5 (𝜃 ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴))
bnj1173.9 ((𝜑𝜓) → 𝑅 FrSe 𝐴)
bnj1173.17 ((𝜑𝜓) → 𝑋𝐴)
Assertion
Ref Expression
bnj1173 ((𝜑𝜓𝑧𝐶) → (𝜃𝑤𝐴))

Proof of Theorem bnj1173
StepHypRef Expression
1 bnj1173.5 . . 3 (𝜃 ↔ ((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴))
2 3simpc 1147 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴) → ((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴))
3 bnj1173.9 . . . . . . 7 ((𝜑𝜓) → 𝑅 FrSe 𝐴)
433adant3 1129 . . . . . 6 ((𝜑𝜓𝑧𝐶) → 𝑅 FrSe 𝐴)
5 bnj1173.17 . . . . . . 7 ((𝜑𝜓) → 𝑋𝐴)
653adant3 1129 . . . . . 6 ((𝜑𝜓𝑧𝐶) → 𝑋𝐴)
7 elin 3900 . . . . . . . . 9 (𝑧 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ↔ (𝑧 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧𝐵))
87simplbi 501 . . . . . . . 8 (𝑧 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
9 bnj1173.3 . . . . . . . 8 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)
108, 9eleq2s 2911 . . . . . . 7 (𝑧𝐶𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
11103ad2ant3 1132 . . . . . 6 ((𝜑𝜓𝑧𝐶) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
12 pm3.21 475 . . . . . 6 ((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) → (((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴) → (((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴) ∧ (𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))))
134, 6, 11, 12syl3anc 1368 . . . . 5 ((𝜑𝜓𝑧𝐶) → (((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴) → (((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴) ∧ (𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))))
14 bnj170 32076 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴) ↔ (((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴) ∧ (𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))))
1513, 14syl6ibr 255 . . . 4 ((𝜑𝜓𝑧𝐶) → (((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴) → ((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴)))
162, 15impbid2 229 . . 3 ((𝜑𝜓𝑧𝐶) → (((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴) ↔ ((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴)))
171, 16syl5bb 286 . 2 ((𝜑𝜓𝑧𝐶) → (𝜃 ↔ ((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴)))
18 bnj1147 32374 . . . . 5 trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴
1918, 11bnj1213 32178 . . . 4 ((𝜑𝜓𝑧𝐶) → 𝑧𝐴)
204, 19jca 515 . . 3 ((𝜑𝜓𝑧𝐶) → (𝑅 FrSe 𝐴𝑧𝐴))
2120biantrurd 536 . 2 ((𝜑𝜓𝑧𝐶) → (𝑤𝐴 ↔ ((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴)))
2217, 21bitr4d 285 1 ((𝜑𝜓𝑧𝐶) → (𝜃𝑤𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  cin 3883   FrSe w-bnj15 32070   trClc-bnj18 32072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-tr 5140  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fn 6331  df-fv 6336  df-om 7565  df-bnj17 32065  df-bnj14 32067  df-bnj18 32073
This theorem is referenced by:  bnj1190  32388
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