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Theorem 3dimlem1 39461
Description: Lemma for 3dim1 39470. (Contributed by NM, 25-Jul-2012.)
Hypotheses
Ref Expression
3dim0.j = (join‘𝐾)
3dim0.l = (le‘𝐾)
3dim0.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
3dimlem1 (((𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅) ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆)) ∧ 𝑃 = 𝑄) → (𝑃𝑅 ∧ ¬ 𝑆 (𝑃 𝑅) ∧ ¬ 𝑇 ((𝑃 𝑅) 𝑆)))

Proof of Theorem 3dimlem1
StepHypRef Expression
1 neeq1 3002 . . 3 (𝑃 = 𝑄 → (𝑃𝑅𝑄𝑅))
2 oveq1 7439 . . . . 5 (𝑃 = 𝑄 → (𝑃 𝑅) = (𝑄 𝑅))
32breq2d 5154 . . . 4 (𝑃 = 𝑄 → (𝑆 (𝑃 𝑅) ↔ 𝑆 (𝑄 𝑅)))
43notbid 318 . . 3 (𝑃 = 𝑄 → (¬ 𝑆 (𝑃 𝑅) ↔ ¬ 𝑆 (𝑄 𝑅)))
52oveq1d 7447 . . . . 5 (𝑃 = 𝑄 → ((𝑃 𝑅) 𝑆) = ((𝑄 𝑅) 𝑆))
65breq2d 5154 . . . 4 (𝑃 = 𝑄 → (𝑇 ((𝑃 𝑅) 𝑆) ↔ 𝑇 ((𝑄 𝑅) 𝑆)))
76notbid 318 . . 3 (𝑃 = 𝑄 → (¬ 𝑇 ((𝑃 𝑅) 𝑆) ↔ ¬ 𝑇 ((𝑄 𝑅) 𝑆)))
81, 4, 73anbi123d 1437 . 2 (𝑃 = 𝑄 → ((𝑃𝑅 ∧ ¬ 𝑆 (𝑃 𝑅) ∧ ¬ 𝑇 ((𝑃 𝑅) 𝑆)) ↔ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅) ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))))
98biimparc 479 1 (((𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅) ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆)) ∧ 𝑃 = 𝑄) → (𝑃𝑅 ∧ ¬ 𝑆 (𝑃 𝑅) ∧ ¬ 𝑇 ((𝑃 𝑅) 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086   = wceq 1539  wne 2939   class class class wbr 5142  cfv 6560  (class class class)co 7432  lecple 17305  joincjn 18358  Atomscatm 39265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ov 7435
This theorem is referenced by:  3dim1  39470
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