MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  raltpg Structured version   Visualization version   GIF version

Theorem raltpg 4633
Description: Convert a restricted universal 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 4631 . . . 4 ((𝐴𝑉𝐵𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓𝜒)))
4 raltpg.3 . . . . 5 (𝑥 = 𝐶 → (𝜑𝜃))
54ralsng 4612 . . . 4 (𝐶𝑋 → (∀𝑥 ∈ {𝐶}𝜑𝜃))
63, 5bi2anan9 635 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐶𝑋) → ((∀𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ∀𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∧ 𝜃)))
763impa 1104 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((∀𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ∀𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓𝜒) ∧ 𝜃)))
8 df-tp 4569 . . . 4 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
98raleqi 3419 . . 3 (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ ∀𝑥 ∈ ({𝐴, 𝐵} ∪ {𝐶})𝜑)
10 ralunb 4171 . . 3 (∀𝑥 ∈ ({𝐴, 𝐵} ∪ {𝐶})𝜑 ↔ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ∀𝑥 ∈ {𝐶}𝜑))
119, 10bitri 276 . 2 (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ∀𝑥 ∈ {𝐶}𝜑))
12 df-3an 1083 . 2 ((𝜓𝜒𝜃) ↔ ((𝜓𝜒) ∧ 𝜃))
137, 11, 123bitr4g 315 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓𝜒𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wral 3143  cun 3938  {csn 4564  {cpr 4566  {ctp 4568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-v 3502  df-sbc 3777  df-un 3945  df-sn 4565  df-pr 4567  df-tp 4569
This theorem is referenced by:  raltp  4640  raltpd  4715  f13dfv  7025  sumtp  15094  lcmftp  15970  nb3grpr  27078  frgr3v  27968
  Copyright terms: Public domain W3C validator