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Theorem 3dimlem1 37950
Description: Lemma for 3dim1 37959. (Contributed by NM, 25-Jul-2012.)
Hypotheses
Ref Expression
3dim0.j ∨ = (joinβ€˜πΎ)
3dim0.l ≀ = (leβ€˜πΎ)
3dim0.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
3dimlem1 (((𝑄 β‰  𝑅 ∧ Β¬ 𝑆 ≀ (𝑄 ∨ 𝑅) ∧ Β¬ 𝑇 ≀ ((𝑄 ∨ 𝑅) ∨ 𝑆)) ∧ 𝑃 = 𝑄) β†’ (𝑃 β‰  𝑅 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑅) ∧ Β¬ 𝑇 ≀ ((𝑃 ∨ 𝑅) ∨ 𝑆)))

Proof of Theorem 3dimlem1
StepHypRef Expression
1 neeq1 3007 . . 3 (𝑃 = 𝑄 β†’ (𝑃 β‰  𝑅 ↔ 𝑄 β‰  𝑅))
2 oveq1 7369 . . . . 5 (𝑃 = 𝑄 β†’ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
32breq2d 5122 . . . 4 (𝑃 = 𝑄 β†’ (𝑆 ≀ (𝑃 ∨ 𝑅) ↔ 𝑆 ≀ (𝑄 ∨ 𝑅)))
43notbid 318 . . 3 (𝑃 = 𝑄 β†’ (Β¬ 𝑆 ≀ (𝑃 ∨ 𝑅) ↔ Β¬ 𝑆 ≀ (𝑄 ∨ 𝑅)))
52oveq1d 7377 . . . . 5 (𝑃 = 𝑄 β†’ ((𝑃 ∨ 𝑅) ∨ 𝑆) = ((𝑄 ∨ 𝑅) ∨ 𝑆))
65breq2d 5122 . . . 4 (𝑃 = 𝑄 β†’ (𝑇 ≀ ((𝑃 ∨ 𝑅) ∨ 𝑆) ↔ 𝑇 ≀ ((𝑄 ∨ 𝑅) ∨ 𝑆)))
76notbid 318 . . 3 (𝑃 = 𝑄 β†’ (Β¬ 𝑇 ≀ ((𝑃 ∨ 𝑅) ∨ 𝑆) ↔ Β¬ 𝑇 ≀ ((𝑄 ∨ 𝑅) ∨ 𝑆)))
81, 4, 73anbi123d 1437 . 2 (𝑃 = 𝑄 β†’ ((𝑃 β‰  𝑅 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑅) ∧ Β¬ 𝑇 ≀ ((𝑃 ∨ 𝑅) ∨ 𝑆)) ↔ (𝑄 β‰  𝑅 ∧ Β¬ 𝑆 ≀ (𝑄 ∨ 𝑅) ∧ Β¬ 𝑇 ≀ ((𝑄 ∨ 𝑅) ∨ 𝑆))))
98biimparc 481 1 (((𝑄 β‰  𝑅 ∧ Β¬ 𝑆 ≀ (𝑄 ∨ 𝑅) ∧ Β¬ 𝑇 ≀ ((𝑄 ∨ 𝑅) ∨ 𝑆)) ∧ 𝑃 = 𝑄) β†’ (𝑃 β‰  𝑅 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑅) ∧ Β¬ 𝑇 ≀ ((𝑃 ∨ 𝑅) ∨ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   β‰  wne 2944   class class class wbr 5110  β€˜cfv 6501  (class class class)co 7362  lecple 17147  joincjn 18207  Atomscatm 37754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-iota 6453  df-fv 6509  df-ov 7365
This theorem is referenced by:  3dim1  37959
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