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Theorem raltpd 4458
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 1594 . . . . . 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 450 . . . . . . . 8 (𝑥 = 𝐴 → (𝜑 → (𝜓𝜒)))
87pm5.32d 674 . . . . . . 7 (𝑥 = 𝐴 → ((𝜑𝜓) ↔ (𝜑𝜒)))
9 ralprd.2 . . . . . . . . 9 ((𝜑𝑥 = 𝐵) → (𝜓𝜃))
109expcom 450 . . . . . . . 8 (𝑥 = 𝐵 → (𝜑 → (𝜓𝜃)))
1110pm5.32d 674 . . . . . . 7 (𝑥 = 𝐵 → ((𝜑𝜓) ↔ (𝜑𝜃)))
12 raltpd.3 . . . . . . . . 9 ((𝜑𝑥 = 𝐶) → (𝜓𝜏))
1312expcom 450 . . . . . . . 8 (𝑥 = 𝐶 → (𝜑 → (𝜓𝜏)))
1413pm5.32d 674 . . . . . . 7 (𝑥 = 𝐶 → ((𝜑𝜓) ↔ (𝜑𝜏)))
158, 11, 14raltpg 4380 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝜑𝜓) ↔ ((𝜑𝜒) ∧ (𝜑𝜃) ∧ (𝜑𝜏))))
163, 4, 5, 15syl3anc 1477 . . . . 5 (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶} (𝜑𝜓) ↔ ((𝜑𝜒) ∧ (𝜑𝜃) ∧ (𝜑𝜏))))
173tpnzd 4457 . . . . . 6 (𝜑 → {𝐴, 𝐵, 𝐶} ≠ ∅)
18 r19.28zv 4210 . . . . . 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 960 . . 3 (𝜑 → ((𝜑 ∧ (𝜒𝜃𝜏)) ↔ ∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓))
2221bicomd 213 . 2 (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓 ↔ (𝜑 ∧ (𝜒𝜃𝜏))))
2322bianabs 960 1 (𝜑 → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜓 ↔ (𝜒𝜃𝜏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wral 3050  c0 4058  {ctp 4325
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-3or 1073  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-ne 2933  df-ral 3055  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-nul 4059  df-sn 4322  df-pr 4324  df-tp 4326
This theorem is referenced by:  eqwrds3  13905  trgcgrg  25609  tgcgr4  25625  cplgr3v  26541
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