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Theorem raltpd 4306
Description: Convert a quantification over a 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 1443 . . . . . 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 451 . . . . . . . 8 (𝑥 = 𝐴 → (𝜑 → (𝜓𝜒)))
87pm5.32d 670 . . . . . . 7 (𝑥 = 𝐴 → ((𝜑𝜓) ↔ (𝜑𝜒)))
9 ralprd.2 . . . . . . . . 9 ((𝜑𝑥 = 𝐵) → (𝜓𝜃))
109expcom 451 . . . . . . . 8 (𝑥 = 𝐵 → (𝜑 → (𝜓𝜃)))
1110pm5.32d 670 . . . . . . 7 (𝑥 = 𝐵 → ((𝜑𝜓) ↔ (𝜑𝜃)))
12 raltpd.3 . . . . . . . . 9 ((𝜑𝑥 = 𝐶) → (𝜓𝜏))
1312expcom 451 . . . . . . . 8 (𝑥 = 𝐶 → (𝜑 → (𝜓𝜏)))
1413pm5.32d 670 . . . . . . 7 (𝑥 = 𝐶 → ((𝜑𝜓) ↔ (𝜑𝜏)))
158, 11, 14raltpg 4227 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝜑𝜓) ↔ ((𝜑𝜒) ∧ (𝜑𝜃) ∧ (𝜑𝜏))))
163, 4, 5, 15syl3anc 1324 . . . . 5 (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝜑𝜓) ↔ ((𝜑𝜒) ∧ (𝜑𝜃) ∧ (𝜑𝜏))))
173tpnzd 4305 . . . . . 6 (𝜑 → {𝐴, 𝐵, 𝐶} ≠ ∅)
18 r19.28zv 4057 . . . . . 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 296 . . . 4 (𝜑 → ((𝜑 ∧ (𝜒𝜃𝜏)) ↔ (𝜑 ∧ ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓)))
2120bianabs 923 . . 3 (𝜑 → ((𝜑 ∧ (𝜒𝜃𝜏)) ↔ ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓))
2221bicomd 213 . 2 (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓 ↔ (𝜑 ∧ (𝜒𝜃𝜏))))
2322bianabs 923 1 (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓 ↔ (𝜒𝜃𝜏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  wne 2791  wral 2909  c0 3907  {ctp 4172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-nul 3908  df-sn 4169  df-pr 4171  df-tp 4173
This theorem is referenced by:  eqwrds3  13685  trgcgrg  25391  tgcgr4  25407  cplgr3v  26312
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