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Theorem 3dimlem1 35479
Description: Lemma for 3dim1 35488. (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 3033 . . 3 (𝑃 = 𝑄 → (𝑃𝑅𝑄𝑅))
2 oveq1 6885 . . . . 5 (𝑃 = 𝑄 → (𝑃 𝑅) = (𝑄 𝑅))
32breq2d 4855 . . . 4 (𝑃 = 𝑄 → (𝑆 (𝑃 𝑅) ↔ 𝑆 (𝑄 𝑅)))
43notbid 310 . . 3 (𝑃 = 𝑄 → (¬ 𝑆 (𝑃 𝑅) ↔ ¬ 𝑆 (𝑄 𝑅)))
52oveq1d 6893 . . . . 5 (𝑃 = 𝑄 → ((𝑃 𝑅) 𝑆) = ((𝑄 𝑅) 𝑆))
65breq2d 4855 . . . 4 (𝑃 = 𝑄 → (𝑇 ((𝑃 𝑅) 𝑆) ↔ 𝑇 ((𝑄 𝑅) 𝑆)))
76notbid 310 . . 3 (𝑃 = 𝑄 → (¬ 𝑇 ((𝑃 𝑅) 𝑆) ↔ ¬ 𝑇 ((𝑄 𝑅) 𝑆)))
81, 4, 73anbi123d 1561 . 2 (𝑃 = 𝑄 → ((𝑃𝑅 ∧ ¬ 𝑆 (𝑃 𝑅) ∧ ¬ 𝑇 ((𝑃 𝑅) 𝑆)) ↔ (𝑄𝑅 ∧ ¬ 𝑆 (𝑄 𝑅) ∧ ¬ 𝑇 ((𝑄 𝑅) 𝑆))))
98biimparc 472 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 385  w3a 1108   = wceq 1653  wne 2971   class class class wbr 4843  cfv 6101  (class class class)co 6878  lecple 16274  joincjn 17259  Atomscatm 35284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-iota 6064  df-fv 6109  df-ov 6881
This theorem is referenced by:  3dim1  35488
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