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Theorem bnj1173 30324
 Description: Technical lemma for bnj69 30332. 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 1053 . . . 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 1074 . . . . . 6 ((𝜑𝜓𝑧𝐶) → 𝑅 FrSe 𝐴)
5 bnj1173.17 . . . . . . 7 ((𝜑𝜓) → 𝑋𝐴)
653adant3 1074 . . . . . 6 ((𝜑𝜓𝑧𝐶) → 𝑋𝐴)
7 elin 3758 . . . . . . . . 9 (𝑧 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ↔ (𝑧 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧𝐵))
87simplbi 475 . . . . . . . 8 (𝑧 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
9 bnj1173.3 . . . . . . . 8 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)
108, 9eleq2s 2706 . . . . . . 7 (𝑧𝐶𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
11103ad2ant3 1077 . . . . . 6 ((𝜑𝜓𝑧𝐶) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
12 pm3.21 463 . . . . . 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 1318 . . . . 5 ((𝜑𝜓𝑧𝐶) → (((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴) → (((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴) ∧ (𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))))
14 bnj170 30017 . . . . 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 241 . . . 4 ((𝜑𝜓𝑧𝐶) → (((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴) → ((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴)))
162, 15impbid2 215 . . 3 ((𝜑𝜓𝑧𝐶) → (((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴) ↔ ((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴)))
171, 16syl5bb 271 . 2 ((𝜑𝜓𝑧𝐶) → (𝜃 ↔ ((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴)))
18 bnj1147 30316 . . . . 5 trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴
1918, 11bnj1213 30123 . . . 4 ((𝜑𝜓𝑧𝐶) → 𝑧𝐴)
204, 19jca 553 . . 3 ((𝜑𝜓𝑧𝐶) → (𝑅 FrSe 𝐴𝑧𝐴))
2120biantrurd 528 . 2 ((𝜑𝜓𝑧𝐶) → (𝑤𝐴 ↔ ((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴)))
2217, 21bitr4d 270 1 ((𝜑𝜓𝑧𝐶) → (𝜃𝑤𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∩ cin 3539   FrSe w-bnj15 30011   trClc-bnj18 30013 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fn 5807  df-fv 5812  df-om 6958  df-bnj17 30006  df-bnj14 30008  df-bnj18 30014 This theorem is referenced by:  bnj1190  30330
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