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Theorem raltpg 4392
Description: Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
ralprg.1 (𝑥 = 𝐴 → (𝜑𝜓))
ralprg.2 (𝑥 = 𝐵 → (𝜑𝜒))
raltpg.3 (𝑥 = 𝐶 → (𝜑𝜃))
Assertion
Ref Expression
raltpg ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜓,𝑥   𝜒,𝑥   𝜃,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑋(𝑥)

Proof of Theorem raltpg
StepHypRef Expression
1 ralprg.1 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
2 ralprg.2 . . . . 5 (𝑥 = 𝐵 → (𝜑𝜒))
31, 2ralprg 4390 . . . 4 ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
4 raltpg.3 . . . . 5 (𝑥 = 𝐶 → (𝜑𝜃))
54ralsng 4375 . . . 4 (𝐶𝑋 → (∀𝑥 ∈ {𝐶}𝜑𝜃))
63, 5bi2anan9 629 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐶𝑋) → ((∀𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ∀𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∧ 𝜃)))
763impa 1136 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((∀𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ∀𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∧ 𝜃)))
8 df-tp 4339 . . . 4 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
98raleqi 3290 . . 3 (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ ∀𝑥 ∈ ({𝐴, 𝐵} ∪ {𝐶})𝜑)
10 ralunb 3956 . . 3 (∀𝑥 ∈ ({𝐴, 𝐵} ∪ {𝐶})𝜑 ↔ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ∀𝑥 ∈ {𝐶}𝜑))
119, 10bitri 266 . 2 (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ∀𝑥 ∈ {𝐶}𝜑))
12 df-3an 1109 . 2 ((𝜓𝜒𝜃) ↔ ((𝜓𝜒) ∧ 𝜃))
137, 11, 123bitr4g 305 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  cun 3730  {csn 4334  {cpr 4336  {ctp 4338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-v 3352  df-sbc 3597  df-un 3737  df-sn 4335  df-pr 4337  df-tp 4339
This theorem is referenced by:  raltp  4396  raltpd  4468  f13dfv  6722  sumtp  14765  lcmftp  15632  nb3grpr  26563  frgr3v  27513
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