Theorem raltp 4705

Description: Convert a universal quantification over an unordered 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 → (𝜑 ↔ 𝜓))
raltp.5	⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))
raltp.6	⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜃))

Assertion

Ref	Expression
raltp	⊢ (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜓,𝑥   𝜒,𝑥   𝜃,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem raltp

Step	Hyp	Ref	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 → (𝜑 ↔ 𝜓))
5		raltp.5	. . 3 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))
6		raltp.6	. . 3 ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜃))
7	4, 5, 6	raltpg 4698	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)))
8	1, 2, 3, 7	mp3an 1463	1 ⊢ (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  Vcvv 3480  {ctp 4630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-v 3482  df-un 3956  df-sn 4627  df-pr 4629  df-tp 4631

This theorem is referenced by:  fztpval  13626  2wlkdlem4  29948  2pthdlem1  29950  3wlkdlem5  30182  3wlkdlem10  30188  upgr3v3e3cycl  30199  poimirlem9  37636  cycl3grtrilem  47913  usgrexmpl2lem  47985  usgrexmpl2trifr  47996  gpg5nbgrvtx03star  48036  gpg5nbgr3star  48037