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Theorem raltpd 4786
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 1483 . . . . . 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 415 . . . . . . . 8 (𝑥 = 𝐴 → (𝜑 → (𝜓𝜒)))
87pm5.32d 578 . . . . . . 7 (𝑥 = 𝐴 → ((𝜑𝜓) ↔ (𝜑𝜒)))
9 ralprd.2 . . . . . . . . 9 ((𝜑𝑥 = 𝐵) → (𝜓𝜃))
109expcom 415 . . . . . . . 8 (𝑥 = 𝐵 → (𝜑 → (𝜓𝜃)))
1110pm5.32d 578 . . . . . . 7 (𝑥 = 𝐵 → ((𝜑𝜓) ↔ (𝜑𝜃)))
12 raltpd.3 . . . . . . . . 9 ((𝜑𝑥 = 𝐶) → (𝜓𝜏))
1312expcom 415 . . . . . . . 8 (𝑥 = 𝐶 → (𝜑 → (𝜓𝜏)))
1413pm5.32d 578 . . . . . . 7 (𝑥 = 𝐶 → ((𝜑𝜓) ↔ (𝜑𝜏)))
158, 11, 14raltpg 4703 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝜑𝜓) ↔ ((𝜑𝜒) ∧ (𝜑𝜃) ∧ (𝜑𝜏))))
163, 4, 5, 15syl3anc 1372 . . . . 5 (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝜑𝜓) ↔ ((𝜑𝜒) ∧ (𝜑𝜃) ∧ (𝜑𝜏))))
173tpnzd 4785 . . . . . 6 (𝜑 → {𝐴, 𝐵, 𝐶} ≠ ∅)
18 r19.28zv 4501 . . . . . 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 307 . . . 4 (𝜑 → ((𝜑 ∧ (𝜒𝜃𝜏)) ↔ (𝜑 ∧ ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓)))
2120bianabs 543 . . 3 (𝜑 → ((𝜑 ∧ (𝜒𝜃𝜏)) ↔ ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓))
2221bicomd 222 . 2 (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓 ↔ (𝜑 ∧ (𝜒𝜃𝜏))))
2322bianabs 543 1 (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓 ↔ (𝜒𝜃𝜏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2941  wral 3062  c0 4323  {ctp 4633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-v 3477  df-dif 3952  df-un 3954  df-nul 4324  df-sn 4630  df-pr 4632  df-tp 4634
This theorem is referenced by:  eqwrds3  14912  trgcgrg  27766  tgcgr4  27782  cplgr3v  28692
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