Theorem raltp 4667

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 4660	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)))
8	1, 2, 3, 7	mp3an 1491	1 ⊢ (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454  {ctp 4592

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-v 3456  df-un 4154  df-sn 4589  df-pr 4591  df-tp 4593

This theorem is used by:  fztpval  13760  2wlkdlem4  30574  2pthdlem1  30576  3wlkdlem5  30821  3wlkdlem10  30827  upgr3v3e3cycl  30838  poimirlem9  38815  cycl3grtrilem  49317  usgrexmpl2lem  49397  usgrexmpl2trifr  49408  gpg5nbgrvtx03star  49451  gpg5nbgr3star  49452