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Theorem meetval2lem 17502
Description: Lemma for meetval2 17503 and meeteu 17504. (Contributed by NM, 12-Sep-2018.) TODO: combine this through meeteu 17504 into meetlem 17505?
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 4929 . . 3 (𝑦 = 𝑋 → (𝑥 𝑦𝑥 𝑋))
2 breq2 4929 . . 3 (𝑦 = 𝑌 → (𝑥 𝑦𝑥 𝑌))
31, 2ralprg 4502 . 2 ((𝑋𝐵𝑌𝐵) → (∀𝑦 ∈ {𝑋, 𝑌}𝑥 𝑦 ↔ (𝑥 𝑋𝑥 𝑌)))
4 breq2 4929 . . . . 5 (𝑦 = 𝑋 → (𝑧 𝑦𝑧 𝑋))
5 breq2 4929 . . . . 5 (𝑦 = 𝑌 → (𝑧 𝑦𝑧 𝑌))
64, 5ralprg 4502 . . . 4 ((𝑋𝐵𝑌𝐵) → (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦 ↔ (𝑧 𝑋𝑧 𝑌)))
76imbi1d 334 . . 3 ((𝑋𝐵𝑌𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥) ↔ ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥)))
87ralbidv 3140 . 2 ((𝑋𝐵𝑌𝐵) → (∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥) ↔ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥)))
93, 8anbi12d 622 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 198  wa 387   = wceq 1508  wcel 2051  wral 3081  {cpr 4437   class class class wbr 4925  cfv 6185  Basecbs 16337  lecple 16426  meetcmee 17425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4926
This theorem is referenced by:  meetval2  17503  meeteu  17504
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