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Theorem raltp 4378
Description: Convert a quantification over a triple to a conjunction. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
raltp.1 𝐴 ∈ V
raltp.2 𝐵 ∈ V
raltp.3 𝐶 ∈ V
raltp.4 (𝑥 = 𝐴 → (𝜑𝜓))
raltp.5 (𝑥 = 𝐵 → (𝜑𝜒))
raltp.6 (𝑥 = 𝐶 → (𝜑𝜃))
Assertion
Ref Expression
raltp (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜓,𝑥   𝜒,𝑥   𝜃,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem raltp
StepHypRef Expression
1 raltp.1 . 2 𝐴 ∈ V
2 raltp.2 . 2 𝐵 ∈ V
3 raltp.3 . 2 𝐶 ∈ V
4 raltp.4 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
5 raltp.5 . . 3 (𝑥 = 𝐵 → (𝜑𝜒))
6 raltp.6 . . 3 (𝑥 = 𝐶 → (𝜑𝜃))
74, 5, 6raltpg 4374 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃)))
81, 2, 3, 7mp3an 1572 1 (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1071   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  {ctp 4321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-v 3353  df-sbc 3588  df-un 3728  df-sn 4318  df-pr 4320  df-tp 4322
This theorem is referenced by:  fztpval  12609  2wlkdlem4  27075  2pthdlem1  27077  3wlkdlem5  27343  3wlkdlem10  27349  upgr3v3e3cycl  27360  poimirlem9  33751
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