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Theorem joinval2lem 17055
Description: Lemma for joinval2 17056 and joineu 17057. (Contributed by NM, 12-Sep-2018.) TODO: combine this through joineu into joinlem?
Hypotheses
Ref Expression
joinval2.b 𝐵 = (Base‘𝐾)
joinval2.l = (le‘𝐾)
joinval2.j = (join‘𝐾)
joinval2.k (𝜑𝐾𝑉)
joinval2.x (𝜑𝑋𝐵)
joinval2.y (𝜑𝑌𝐵)
Assertion
Ref Expression
joinval2lem ((𝑋𝐵𝑌𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧𝑥 𝑧)) ↔ ((𝑋 𝑥𝑌 𝑥) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → 𝑥 𝑧))))
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)   (𝑦)   (𝑥,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem joinval2lem
StepHypRef Expression
1 breq1 4688 . . 3 (𝑦 = 𝑋 → (𝑦 𝑥𝑋 𝑥))
2 breq1 4688 . . 3 (𝑦 = 𝑌 → (𝑦 𝑥𝑌 𝑥))
31, 2ralprg 4266 . 2 ((𝑋𝐵𝑌𝐵) → (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑥 ↔ (𝑋 𝑥𝑌 𝑥)))
4 breq1 4688 . . . . 5 (𝑦 = 𝑋 → (𝑦 𝑧𝑋 𝑧))
5 breq1 4688 . . . . 5 (𝑦 = 𝑌 → (𝑦 𝑧𝑌 𝑧))
64, 5ralprg 4266 . . . 4 ((𝑋𝐵𝑌𝐵) → (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧 ↔ (𝑋 𝑧𝑌 𝑧)))
76imbi1d 330 . . 3 ((𝑋𝐵𝑌𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧𝑥 𝑧) ↔ ((𝑋 𝑧𝑌 𝑧) → 𝑥 𝑧)))
87ralbidv 3015 . 2 ((𝑋𝐵𝑌𝐵) → (∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧𝑥 𝑧) ↔ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → 𝑥 𝑧)))
93, 8anbi12d 747 1 ((𝑋𝐵𝑌𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧𝑥 𝑧)) ↔ ((𝑋 𝑥𝑌 𝑥) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → 𝑥 𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  {cpr 4212   class class class wbr 4685  cfv 5926  Basecbs 15904  lecple 15995  joincjn 16991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686
This theorem is referenced by:  joinval2  17056  joineu  17057
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