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Theorem meetval2lem 17900
Description: Lemma for meetval2 17901 and meeteu 17902. (Contributed by NM, 12-Sep-2018.) TODO: combine this through meeteu 17902 into meetlem 17903?
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 5057 . . 3 (𝑦 = 𝑋 → (𝑥 𝑦𝑥 𝑋))
2 breq2 5057 . . 3 (𝑦 = 𝑌 → (𝑥 𝑦𝑥 𝑌))
31, 2ralprg 4610 . 2 ((𝑋𝐵𝑌𝐵) → (∀𝑦 ∈ {𝑋, 𝑌}𝑥 𝑦 ↔ (𝑥 𝑋𝑥 𝑌)))
4 breq2 5057 . . . . 5 (𝑦 = 𝑋 → (𝑧 𝑦𝑧 𝑋))
5 breq2 5057 . . . . 5 (𝑦 = 𝑌 → (𝑧 𝑦𝑧 𝑌))
64, 5ralprg 4610 . . . 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 3118 . 2 ((𝑋𝐵𝑌𝐵) → (∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥) ↔ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥)))
93, 8anbi12d 634 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 1543  wcel 2110  wral 3061  {cpr 4543   class class class wbr 5053  cfv 6380  Basecbs 16760  lecple 16809  meetcmee 17819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054
This theorem is referenced by:  meetval2  17901  meeteu  17902  meetdm3  45938
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