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Theorem bnj1173 34541
Description: Technical lemma for bnj69 34549. 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 3959 . . . . . . . . 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 2845 . . . . . . 7 (𝑧𝐶𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
11103ad2ant3 1132 . . . . . 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 1368 . . . . 5 ((𝜑𝜓𝑧𝐶) → (((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴) → (((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴) ∧ (𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))))
14 bnj170 34237 . . . . 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 251 . . . 4 ((𝜑𝜓𝑧𝐶) → (((𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴) → ((𝑅 FrSe 𝐴𝑋𝐴𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ∧ (𝑅 FrSe 𝐴𝑧𝐴) ∧ 𝑤𝐴)))
162, 15impbid2 225 . . 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 34533 . . . . 5 trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐴
1918, 11bnj1213 34337 . . . 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 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  cin 3942   FrSe w-bnj15 34231   trClc-bnj18 34233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fn 6539  df-fv 6544  df-om 7852  df-bnj17 34226  df-bnj14 34228  df-bnj18 34234
This theorem is referenced by:  bnj1190  34547
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