Theorem 2ralunsn 3824

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 3823	. . 3 ⊢ (𝐵 ∈ 𝐶 → (∀𝑦 ∈ (𝐴 ∪ {𝐵})𝜑 ↔ (∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓)))
3	2	ralbidv 2494	. 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 2494	. . . . 5 ⊢ (𝑥 = 𝐵 → (∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜒))
6		2ralunsn.3	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜃))
7	5, 6	anbi12d 473	. . . 4 ⊢ (𝑥 = 𝐵 → ((∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓) ↔ (∀𝑦 ∈ 𝐴 𝜒 ∧ 𝜃)))
8	7	ralunsn 3823	. . 3 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ (𝐴 ∪ {𝐵})(∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓) ∧ (∀𝑦 ∈ 𝐴 𝜒 ∧ 𝜃))))
9		r19.26 2620	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))
10	9	anbi1i 458	. . 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 196	. 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 188	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   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164  ∀wral 2472   ∪ cun 3151  {csn 3618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-sbc 2986  df-un 3157  df-sn 3624

This theorem is referenced by: (None)