Theorem raltp 4710

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 4703	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)))
8	1, 2, 3, 7	mp3an 1460	1 ⊢ (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059  Vcvv 3478  {ctp 4635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-v 3480  df-un 3968  df-sn 4632  df-pr 4634  df-tp 4636

This theorem is referenced by:  fztpval  13623  2wlkdlem4  29958  2pthdlem1  29960  3wlkdlem5  30192  3wlkdlem10  30198  upgr3v3e3cycl  30209  poimirlem9  37616  usgrexmpl2lem  47921  usgrexmpl2trifr  47932  gpg5nbgrvtx03star  47971  gpg5nbgr3star  47972