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Theorem bnj1173 35016
Description: Technical lemma for bnj69 35024. 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 1151 . . . 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 1133 . . . . . 6 ((𝜑𝜓𝑧𝐶) → 𝑅 FrSe 𝐴)
5 bnj1173.17 . . . . . . 7 ((𝜑𝜓) → 𝑋𝐴)
653adant3 1133 . . . . . 6 ((𝜑𝜓𝑧𝐶) → 𝑋𝐴)
7 elin 3967 . . . . . . . . 9 (𝑧 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) ↔ (𝑧 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧𝐵))
87simplbi 497 . . . . . . . 8 (𝑧 ∈ ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
9 bnj1173.3 . . . . . . . 8 𝐶 = ( trCl(𝑋, 𝐴, 𝑅) ∩ 𝐵)
108, 9eleq2s 2859 . . . . . . 7 (𝑧𝐶𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
11103ad2ant3 1136 . . . . . 6 ((𝜑𝜓𝑧𝐶) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
12 pm3.21 471 . . . . . 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 1373 . . . . 5 ((𝜑𝜓𝑧𝐶) → (((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴) → (((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴) ∧ (𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))))
14 bnj170 34712 . . . . 5 (((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴) ↔ (((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴) ∧ (𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))))
1513, 14imbitrrdi 252 . . . 4 ((𝜑𝜓𝑧𝐶) → (((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴) → ((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴)))
162, 15impbid2 226 . . 3 ((𝜑𝜓𝑧𝐶) → (((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴) ↔ ((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴)))
171, 16bitrid 283 . 2 ((𝜑𝜓𝑧𝐶) → (𝜃 ↔ ((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴)))
18 bnj1147 35008 . . . . 5 trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴
1918, 11bnj1213 34812 . . . 4 ((𝜑𝜓𝑧𝐶) → 𝑧𝐴)
204, 19jca 511 . . 3 ((𝜑𝜓𝑧𝐶) → (𝑅 FrSe 𝐴𝑧𝐴))
2120biantrurd 532 . 2 ((𝜑𝜓𝑧𝐶) → (𝑤𝐴 ↔ ((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴)))
2217, 21bitr4d 282 1 ((𝜑𝜓𝑧𝐶) → (𝜃𝑤𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  cin 3950   FrSe w-bnj15 34706   trClc-bnj18 34708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fn 6564  df-fv 6569  df-om 7888  df-bnj17 34701  df-bnj14 34703  df-bnj18 34709
This theorem is referenced by:  bnj1190  35022
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