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Theorem meetval2lem 17626
 Description: Lemma for meetval2 17627 and meeteu 17628. (Contributed by NM, 12-Sep-2018.) TODO: combine this through meeteu 17628 into meetlem 17629?
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 5034 . . 3 (𝑦 = 𝑋 → (𝑥 𝑦𝑥 𝑋))
2 breq2 5034 . . 3 (𝑦 = 𝑌 → (𝑥 𝑦𝑥 𝑌))
31, 2ralprg 4592 . 2 ((𝑋𝐵𝑌𝐵) → (∀𝑦 ∈ {𝑋, 𝑌}𝑥 𝑦 ↔ (𝑥 𝑋𝑥 𝑌)))
4 breq2 5034 . . . . 5 (𝑦 = 𝑋 → (𝑧 𝑦𝑧 𝑋))
5 breq2 5034 . . . . 5 (𝑦 = 𝑌 → (𝑧 𝑦𝑧 𝑌))
64, 5ralprg 4592 . . . 4 ((𝑋𝐵𝑌𝐵) → (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦 ↔ (𝑧 𝑋𝑧 𝑌)))
76imbi1d 345 . . 3 ((𝑋𝐵𝑌𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥) ↔ ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥)))
87ralbidv 3162 . 2 ((𝑋𝐵𝑌𝐵) → (∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥) ↔ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥)))
93, 8anbi12d 633 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  {cpr 4527   class class class wbr 5030  ‘cfv 6324  Basecbs 16477  lecple 16566  meetcmee 17549 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-v 3443  df-sbc 3721  df-un 3886  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031 This theorem is referenced by:  meetval2  17627  meeteu  17628
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