Theorem raltpg 4659

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 4657	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒)))
4		raltpg.3	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜃))
5	4	ralsng 4636	. . . 4 ⊢ (𝐶 ∈ 𝑋 → (∀𝑥 ∈ {𝐶}𝜑 ↔ 𝜃))
6	3, 5	bi2anan9 650	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐶 ∈ 𝑋) → ((∀𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ∀𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜃)))
7	6	3impa 1127	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((∀𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ∀𝑥 ∈ {𝐶}𝜑) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜃)))
8		df-tp 4593	. . . 4 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
9	8	raleqi 3321	. . 3 ⊢ (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ ∀𝑥 ∈ ({𝐴, 𝐵} ∪ {𝐶})𝜑)
10		ralunb 4198	. . 3 ⊢ (∀𝑥 ∈ ({𝐴, 𝐵} ∪ {𝐶})𝜑 ↔ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ∀𝑥 ∈ {𝐶}𝜑))
11	9, 10	bitri 278	. 2 ⊢ (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ∧ ∀𝑥 ∈ {𝐶}𝜑))
12		df-3an 1105	. 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜃))
13	7, 11, 12	3bitr4g 317	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∀𝑥 ∈ {𝐴, 𝐵, 𝐶}𝜑 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ∪ cun 4152  {csn 4588  {cpr 4590  {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:  raltp  4666  raltpd  4742  f13dfv  7302  sumtp  15941  lcmftp  16837  nb3grpr  29990  frgr3v  30903  usgrexmpl1lem  49322