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Theorem joinval2lem 17229
Description: Lemma for joinval2 17230 and joineu 17231. (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 4807 . . 3 (𝑦 = 𝑋 → (𝑦 𝑥𝑋 𝑥))
2 breq1 4807 . . 3 (𝑦 = 𝑌 → (𝑦 𝑥𝑌 𝑥))
31, 2ralprg 4378 . 2 ((𝑋𝐵𝑌𝐵) → (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑥 ↔ (𝑋 𝑥𝑌 𝑥)))
4 breq1 4807 . . . . 5 (𝑦 = 𝑋 → (𝑦 𝑧𝑋 𝑧))
5 breq1 4807 . . . . 5 (𝑦 = 𝑌 → (𝑦 𝑧𝑌 𝑧))
64, 5ralprg 4378 . . . 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 3124 . 2 ((𝑋𝐵𝑌𝐵) → (∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧𝑥 𝑧) ↔ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → 𝑥 𝑧)))
93, 8anbi12d 749 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 1632  wcel 2139  wral 3050  {cpr 4323   class class class wbr 4804  cfv 6049  Basecbs 16079  lecple 16170  joincjn 17165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805
This theorem is referenced by:  joinval2  17230  joineu  17231
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