Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  raltpd Structured version   Visualization version   GIF version

Theorem raltpd 4674
 Description: Convert a universal quantification over an unordered triple to a conjunction. (Contributed by Thierry Arnoux, 8-Apr-2019.)
Hypotheses
Ref Expression
ralprd.1 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
ralprd.2 ((𝜑𝑥 = 𝐵) → (𝜓𝜃))
raltpd.3 ((𝜑𝑥 = 𝐶) → (𝜓𝜏))
ralprd.a (𝜑𝐴𝑉)
ralprd.b (𝜑𝐵𝑊)
raltpd.c (𝜑𝐶𝑋)
Assertion
Ref Expression
raltpd (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓 ↔ (𝜒𝜃𝜏)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥   𝜒,𝑥   𝜃,𝑥   𝜏,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑋(𝑥)

Proof of Theorem raltpd
StepHypRef Expression
1 an3andi 1479 . . . . . 6 ((𝜑 ∧ (𝜒𝜃𝜏)) ↔ ((𝜑𝜒) ∧ (𝜑𝜃) ∧ (𝜑𝜏)))
21a1i 11 . . . . 5 (𝜑 → ((𝜑 ∧ (𝜒𝜃𝜏)) ↔ ((𝜑𝜒) ∧ (𝜑𝜃) ∧ (𝜑𝜏))))
3 ralprd.a . . . . . 6 (𝜑𝐴𝑉)
4 ralprd.b . . . . . 6 (𝜑𝐵𝑊)
5 raltpd.c . . . . . 6 (𝜑𝐶𝑋)
6 ralprd.1 . . . . . . . . 9 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
76expcom 417 . . . . . . . 8 (𝑥 = 𝐴 → (𝜑 → (𝜓𝜒)))
87pm5.32d 580 . . . . . . 7 (𝑥 = 𝐴 → ((𝜑𝜓) ↔ (𝜑𝜒)))
9 ralprd.2 . . . . . . . . 9 ((𝜑𝑥 = 𝐵) → (𝜓𝜃))
109expcom 417 . . . . . . . 8 (𝑥 = 𝐵 → (𝜑 → (𝜓𝜃)))
1110pm5.32d 580 . . . . . . 7 (𝑥 = 𝐵 → ((𝜑𝜓) ↔ (𝜑𝜃)))
12 raltpd.3 . . . . . . . . 9 ((𝜑𝑥 = 𝐶) → (𝜓𝜏))
1312expcom 417 . . . . . . . 8 (𝑥 = 𝐶 → (𝜑 → (𝜓𝜏)))
1413pm5.32d 580 . . . . . . 7 (𝑥 = 𝐶 → ((𝜑𝜓) ↔ (𝜑𝜏)))
158, 11, 14raltpg 4591 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝜑𝜓) ↔ ((𝜑𝜒) ∧ (𝜑𝜃) ∧ (𝜑𝜏))))
163, 4, 5, 15syl3anc 1368 . . . . 5 (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝜑𝜓) ↔ ((𝜑𝜒) ∧ (𝜑𝜃) ∧ (𝜑𝜏))))
173tpnzd 4673 . . . . . 6 (𝜑 → {𝐴, 𝐵, 𝐶} ≠ ∅)
18 r19.28zv 4394 . . . . . 6 ({𝐴, 𝐵, 𝐶} ≠ ∅ → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓)))
1917, 18syl 17 . . . . 5 (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓)))
202, 16, 193bitr2d 310 . . . 4 (𝜑 → ((𝜑 ∧ (𝜒𝜃𝜏)) ↔ (𝜑 ∧ ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓)))
2120bianabs 545 . . 3 (𝜑 → ((𝜑 ∧ (𝜒𝜃𝜏)) ↔ ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓))
2221bicomd 226 . 2 (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓 ↔ (𝜑 ∧ (𝜒𝜃𝜏))))
2322bianabs 545 1 (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓 ↔ (𝜒𝜃𝜏)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  ∅c0 4225  {ctp 4526 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 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ne 2952  df-ral 3075  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-nul 4226  df-sn 4523  df-pr 4525  df-tp 4527 This theorem is referenced by:  eqwrds3  14372  trgcgrg  26408  tgcgr4  26424  cplgr3v  27324
 Copyright terms: Public domain W3C validator