Theorem ralprgf 4699

Description: Convert a restricted universal quantification over a pair to a conjunction, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 17-Sep-2011.) (Revised by AV, 8-Apr-2023.)

Hypotheses

Ref	Expression
ralprgf.1	⊢ Ⅎ𝑥𝜓
ralprgf.2	⊢ Ⅎ𝑥𝜒
ralprgf.a	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
ralprgf.b	⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))

Assertion

Ref	Expression
ralprgf	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem ralprgf

Step	Hyp	Ref	Expression
1		df-pr 4633	. . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2	1	raleqi 3319	. . 3 ⊢ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ∀𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑)
3		ralunb 4191	. . 3 ⊢ (∀𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑 ↔ (∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑))
4	2, 3	bitri 274	. 2 ⊢ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑))
5		ralprgf.1	. . . 4 ⊢ Ⅎ𝑥𝜓
6		ralprgf.a	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
7	5, 6	ralsngf 4678	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
8		ralprgf.2	. . . 4 ⊢ Ⅎ𝑥𝜒
9		ralprgf.b	. . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))
10	8, 9	ralsngf 4678	. . 3 ⊢ (𝐵 ∈ 𝑊 → (∀𝑥 ∈ {𝐵}𝜑 ↔ 𝜒))
11	7, 10	bi2anan9 636	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑) ↔ (𝜓 ∧ 𝜒)))
12	4, 11	bitrid 282	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533  Ⅎwnf 1777   ∈ wcel 2098  ∀wral 3057   ∪ cun 3945  {csn 4630  {cpr 4632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2166  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3058  df-rex 3067  df-v 3473  df-sbc 3777  df-un 3952  df-sn 4631  df-pr 4633

This theorem is referenced by:  ralprgOLD  4702