Theorem 2ralunsn 4860

Description: Double restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012.)

Hypotheses

Ref	Expression
2ralunsn.1	⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))
2ralunsn.2	⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓))
2ralunsn.3	⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜃))

Assertion

Ref	Expression
2ralunsn	⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})∀𝑦 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ (∀𝑦 ∈ 𝐴 𝜒 ∧ 𝜃))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶   𝜒,𝑥   𝜓,𝑦   𝜃,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑦)   𝐴(𝑦)   𝐶(𝑦)

Proof of Theorem 2ralunsn

Step	Hyp	Ref	Expression
1		2ralunsn.2	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓))
2	1	ralunsn 4859	. . 3 ⊢ (𝐵 ∈ 𝐶 → (∀𝑦 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓)))
3	2	ralbidv 3188	. 2 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})∀𝑦 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ ∀𝑥 ∈ (𝐴 ∪ {𝐵})(∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓)))
4		2ralunsn.1	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))
5	4	ralbidv 3188	. . . . 5 ⊢ (𝑥 = 𝐵 → (∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜒))
6		2ralunsn.3	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜃))
7	5, 6	anbi12d 643	. . . 4 ⊢ (𝑥 = 𝐵 → ((∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓) ↔ (∀𝑦 ∈ 𝐴 𝜒 ∧ 𝜃)))
8	7	ralunsn 4859	. . 3 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})(∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓) ∧ (∀𝑦 ∈ 𝐴 𝜒 ∧ 𝜃))))
9		r19.26 3125	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))
10	9	anbi1i 635	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓) ∧ (∀𝑦 ∈ 𝐴 𝜒 ∧ 𝜃)) ↔ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ (∀𝑦 ∈ 𝐴 𝜒 ∧ 𝜃)))
11	8, 10	bitrdi 290	. 2 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})(∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓) ↔ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ (∀𝑦 ∈ 𝐴 𝜒 ∧ 𝜃))))
12	3, 11	bitrd 282	1 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})∀𝑦 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ∧ (∀𝑦 ∈ 𝐴 𝜒 ∧ 𝜃))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ∪ cun 3903  {csn 4589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-v 3457  df-un 3910  df-sn 4590

This theorem is referenced by:  disjressuc2  39060