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Theorem meetval2lem 18371
Description: Lemma for meetval2 18372 and meeteu 18373. (Contributed by NM, 12-Sep-2018.) TODO: combine this through meeteu 18373 into meetlem 18374?
Hypotheses
Ref Expression
meetval2.b 𝐡 = (Baseβ€˜πΎ)
meetval2.l ≀ = (leβ€˜πΎ)
meetval2.m ∧ = (meetβ€˜πΎ)
meetval2.k (πœ‘ β†’ 𝐾 ∈ 𝑉)
meetval2.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
meetval2.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
Assertion
Ref Expression
meetval2lem ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((βˆ€π‘¦ ∈ {𝑋, π‘Œ}π‘₯ ≀ 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ {𝑋, π‘Œ}𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯)) ↔ ((π‘₯ ≀ 𝑋 ∧ π‘₯ ≀ π‘Œ) ∧ βˆ€π‘§ ∈ 𝐡 ((𝑧 ≀ 𝑋 ∧ 𝑧 ≀ π‘Œ) β†’ 𝑧 ≀ π‘₯))))
Distinct variable groups:   π‘₯,𝑧,𝐡   π‘₯, ∧ ,𝑧   π‘₯,𝑦,𝐾,𝑧   𝑦, ≀   π‘₯,𝑋,𝑦,𝑧   π‘₯,π‘Œ,𝑦,𝑧
Allowed substitution hints:   πœ‘(π‘₯,𝑦,𝑧)   𝐡(𝑦)   ≀ (π‘₯,𝑧)   ∧ (𝑦)   𝑉(π‘₯,𝑦,𝑧)

Proof of Theorem meetval2lem
StepHypRef Expression
1 breq2 5146 . . 3 (𝑦 = 𝑋 β†’ (π‘₯ ≀ 𝑦 ↔ π‘₯ ≀ 𝑋))
2 breq2 5146 . . 3 (𝑦 = π‘Œ β†’ (π‘₯ ≀ 𝑦 ↔ π‘₯ ≀ π‘Œ))
31, 2ralprg 4694 . 2 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (βˆ€π‘¦ ∈ {𝑋, π‘Œ}π‘₯ ≀ 𝑦 ↔ (π‘₯ ≀ 𝑋 ∧ π‘₯ ≀ π‘Œ)))
4 breq2 5146 . . . . 5 (𝑦 = 𝑋 β†’ (𝑧 ≀ 𝑦 ↔ 𝑧 ≀ 𝑋))
5 breq2 5146 . . . . 5 (𝑦 = π‘Œ β†’ (𝑧 ≀ 𝑦 ↔ 𝑧 ≀ π‘Œ))
64, 5ralprg 4694 . . . 4 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (βˆ€π‘¦ ∈ {𝑋, π‘Œ}𝑧 ≀ 𝑦 ↔ (𝑧 ≀ 𝑋 ∧ 𝑧 ≀ π‘Œ)))
76imbi1d 341 . . 3 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((βˆ€π‘¦ ∈ {𝑋, π‘Œ}𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯) ↔ ((𝑧 ≀ 𝑋 ∧ 𝑧 ≀ π‘Œ) β†’ 𝑧 ≀ π‘₯)))
87ralbidv 3172 . 2 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ {𝑋, π‘Œ}𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯) ↔ βˆ€π‘§ ∈ 𝐡 ((𝑧 ≀ 𝑋 ∧ 𝑧 ≀ π‘Œ) β†’ 𝑧 ≀ π‘₯)))
93, 8anbi12d 630 1 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((βˆ€π‘¦ ∈ {𝑋, π‘Œ}π‘₯ ≀ 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ {𝑋, π‘Œ}𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯)) ↔ ((π‘₯ ≀ 𝑋 ∧ π‘₯ ≀ π‘Œ) ∧ βˆ€π‘§ ∈ 𝐡 ((𝑧 ≀ 𝑋 ∧ 𝑧 ≀ π‘Œ) β†’ 𝑧 ≀ π‘₯))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆ€wral 3056  {cpr 4626   class class class wbr 5142  β€˜cfv 6542  Basecbs 17165  lecple 17225  meetcmee 18289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143
This theorem is referenced by:  meetval2  18372  meeteu  18373  meetdm3  47903
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