Theorem raltpg 4653

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 → (𝜑 ↔ 𝜓))
ralprg.2	⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))
raltpg.3	⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜃))

Assertion

Ref	Expression
raltpg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∀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 hints:   𝜑(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑋(𝑥)

Proof of Theorem raltpg

Step	Hyp	Ref	Expression
1		ralprg.1	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2		ralprg.2	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))
3	1, 2	ralprg 4651	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒)))
4		raltpg.3	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜃))
5	4	ralsng 4630	. . . 4 ⊢ (𝐶 ∈ 𝑋 → (∀𝑥 ∈ {𝐶}𝜑 ↔ 𝜃))
6	3, 5	bi2anan9 638	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐶 ∈ 𝑋) → ((∀𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ∀𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜃)))
7	6	3impa 1109	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((∀𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ∀𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜃)))
8		df-tp 4583	. . . 4 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
9	8	raleqi 3292	. . 3 ⊢ (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ ∀𝑥 ∈ ({𝐴, 𝐵} ∪ {𝐶})𝜑)
10		ralunb 4147	. . 3 ⊢ (∀𝑥 ∈ ({𝐴, 𝐵} ∪ {𝐶})𝜑 ↔ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ∀𝑥 ∈ {𝐶}𝜑))
11	9, 10	bitri 275	. 2 ⊢ (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ∀𝑥 ∈ {𝐶}𝜑))
12		df-3an 1088	. 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜃))
13	7, 11, 12	3bitr4g 314	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3049   ∪ cun 3897  {csn 4578  {cpr 4580  {ctp 4582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rex 3059  df-v 3440  df-un 3904  df-sn 4579  df-pr 4581  df-tp 4583

This theorem is referenced by:  raltp  4660  raltpd  4736  f13dfv  7218  sumtp  15670  lcmftp  16561  nb3grpr  29404  frgr3v  30299  usgrexmpl1lem  48209