Theorem raltp 4240

Description: Convert a quantification over a 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 4236	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)))
8	1, 2, 3, 7	mp3an 1424	1 ⊢ (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃))

Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200  {ctp 4181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-sbc 3436  df-un 3579  df-sn 4178  df-pr 4180  df-tp 4182

This theorem is referenced by:  fztpval  12402  2wlkdlem4  26824  2pthdlem1  26826  3wlkdlem5  27023  3wlkdlem10  27029  upgr3v3e3cycl  27040  poimirlem9  33418