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Theorem cdleme43frv1snN 38900
Description: Value of ⦋𝑅 / π‘ β¦Œπ‘ when Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄). (Contributed by NM, 30-Mar-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemefr27.b 𝐡 = (Baseβ€˜πΎ)
cdlemefr27.l ≀ = (leβ€˜πΎ)
cdlemefr27.j ∨ = (joinβ€˜πΎ)
cdlemefr27.m ∧ = (meetβ€˜πΎ)
cdlemefr27.a 𝐴 = (Atomsβ€˜πΎ)
cdlemefr27.h 𝐻 = (LHypβ€˜πΎ)
cdlemefr27.u π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
cdlemefr27.c 𝐢 = ((𝑠 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ π‘Š)))
cdlemefr27.n 𝑁 = if(𝑠 ≀ (𝑃 ∨ 𝑄), 𝐼, 𝐢)
cdleme43fr.x 𝑋 = ((𝑅 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ π‘Š)))
Assertion
Ref Expression
cdleme43frv1snN ((𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄)) β†’ ⦋𝑅 / π‘ β¦Œπ‘ = 𝑋)
Distinct variable groups:   𝐴,𝑠   ∨ ,𝑠   ≀ ,𝑠   ∧ ,𝑠   𝑃,𝑠   𝑄,𝑠   𝑅,𝑠   π‘ˆ,𝑠   π‘Š,𝑠   𝐻,𝑠   𝐾,𝑠   𝐡,𝑠
Allowed substitution hints:   𝐢(𝑠)   𝐼(𝑠)   𝑁(𝑠)   𝑋(𝑠)

Proof of Theorem cdleme43frv1snN
StepHypRef Expression
1 cdlemefr27.c . 2 𝐢 = ((𝑠 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ π‘Š)))
2 cdlemefr27.n . 2 𝑁 = if(𝑠 ≀ (𝑃 ∨ 𝑄), 𝐼, 𝐢)
3 cdleme43fr.x . 2 𝑋 = ((𝑅 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ π‘Š)))
41, 2, 3cdleme31sn2 38881 1 ((𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄)) β†’ ⦋𝑅 / π‘ β¦Œπ‘ = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  β¦‹csb 3860  ifcif 4491   class class class wbr 5110  β€˜cfv 6501  (class class class)co 7362  Basecbs 17090  lecple 17147  joincjn 18207  meetcmee 18208  Atomscatm 37754  LHypclh 38476
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-10 2138  ax-11 2155  ax-12 2172  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-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  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: (None)
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