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Theorem meetval2lem 18357
Description: Lemma for meetval2 18358 and meeteu 18359. (Contributed by NM, 12-Sep-2018.) TODO: combine this through meeteu 18359 into meetlem 18360?
Hypotheses
Ref Expression
meetval2.b 𝐵 = (Base‘𝐾)
meetval2.l = (le‘𝐾)
meetval2.m = (meet‘𝐾)
meetval2.k (𝜑𝐾𝑉)
meetval2.x (𝜑𝑋𝐵)
meetval2.y (𝜑𝑌𝐵)
Assertion
Ref Expression
meetval2lem ((𝑋𝐵𝑌𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥)) ↔ ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))))
Distinct variable groups:   𝑥,𝑧,𝐵   𝑥, ,𝑧   𝑥,𝑦,𝐾,𝑧   𝑦,   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐵(𝑦)   (𝑥,𝑧)   (𝑦)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem meetval2lem
StepHypRef Expression
1 breq2 5152 . . 3 (𝑦 = 𝑋 → (𝑥 𝑦𝑥 𝑋))
2 breq2 5152 . . 3 (𝑦 = 𝑌 → (𝑥 𝑦𝑥 𝑌))
31, 2ralprg 4698 . 2 ((𝑋𝐵𝑌𝐵) → (∀𝑦 ∈ {𝑋, 𝑌}𝑥 𝑦 ↔ (𝑥 𝑋𝑥 𝑌)))
4 breq2 5152 . . . . 5 (𝑦 = 𝑋 → (𝑧 𝑦𝑧 𝑋))
5 breq2 5152 . . . . 5 (𝑦 = 𝑌 → (𝑧 𝑦𝑧 𝑌))
64, 5ralprg 4698 . . . 4 ((𝑋𝐵𝑌𝐵) → (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦 ↔ (𝑧 𝑋𝑧 𝑌)))
76imbi1d 341 . . 3 ((𝑋𝐵𝑌𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥) ↔ ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥)))
87ralbidv 3176 . 2 ((𝑋𝐵𝑌𝐵) → (∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥) ↔ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥)))
93, 8anbi12d 630 1 ((𝑋𝐵𝑌𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥)) ↔ ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  wral 3060  {cpr 4630   class class class wbr 5148  cfv 6543  Basecbs 17151  lecple 17211  meetcmee 18275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149
This theorem is referenced by:  meetval2  18358  meeteu  18359  meetdm3  47766
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