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Theorem 3dimlem1 38963
Description: Lemma for 3dim1 38972. (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 3000 . . 3 (𝑃 = 𝑄 β†’ (𝑃 β‰  𝑅 ↔ 𝑄 β‰  𝑅))
2 oveq1 7433 . . . . 5 (𝑃 = 𝑄 β†’ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅))
32breq2d 5164 . . . 4 (𝑃 = 𝑄 β†’ (𝑆 ≀ (𝑃 ∨ 𝑅) ↔ 𝑆 ≀ (𝑄 ∨ 𝑅)))
43notbid 317 . . 3 (𝑃 = 𝑄 β†’ (Β¬ 𝑆 ≀ (𝑃 ∨ 𝑅) ↔ Β¬ 𝑆 ≀ (𝑄 ∨ 𝑅)))
52oveq1d 7441 . . . . 5 (𝑃 = 𝑄 β†’ ((𝑃 ∨ 𝑅) ∨ 𝑆) = ((𝑄 ∨ 𝑅) ∨ 𝑆))
65breq2d 5164 . . . 4 (𝑃 = 𝑄 β†’ (𝑇 ≀ ((𝑃 ∨ 𝑅) ∨ 𝑆) ↔ 𝑇 ≀ ((𝑄 ∨ 𝑅) ∨ 𝑆)))
76notbid 317 . . 3 (𝑃 = 𝑄 β†’ (Β¬ 𝑇 ≀ ((𝑃 ∨ 𝑅) ∨ 𝑆) ↔ Β¬ 𝑇 ≀ ((𝑄 ∨ 𝑅) ∨ 𝑆)))
81, 4, 73anbi123d 1432 . 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 478 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 394   ∧ w3a 1084   = wceq 1533   β‰  wne 2937   class class class wbr 5152  β€˜cfv 6553  (class class class)co 7426  lecple 17247  joincjn 18310  Atomscatm 38767
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-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429
This theorem is referenced by:  3dim1  38972
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