Theorem ceqsex2 2681

Description: Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)

Hypotheses

Ref	Expression
ceqsex2.1	⊢ Ⅎ𝑥𝜓
ceqsex2.2	⊢ Ⅎ𝑦𝜒
ceqsex2.3	⊢ 𝐴 ∈ V
ceqsex2.4	⊢ 𝐵 ∈ V
ceqsex2.5	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
ceqsex2.6	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
ceqsex2	⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ 𝜒)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

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

Proof of Theorem ceqsex2

Step	Hyp	Ref	Expression
1		3anass 934	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
2	1	exbii 1552	. . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ ∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
3		19.42v 1845	. . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ (𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)))
4	2, 3	bitri 183	. . 3 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)))
5	4	exbii 1552	. 2 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)))
6		nfv 1476	. . . . 5 ⊢ Ⅎ𝑥 𝑦 = 𝐵
7		ceqsex2.1	. . . . 5 ⊢ Ⅎ𝑥𝜓
8	6, 7	nfan 1512	. . . 4 ⊢ Ⅎ𝑥(𝑦 = 𝐵 ∧ 𝜓)
9	8	nfex 1584	. . 3 ⊢ Ⅎ𝑥∃𝑦(𝑦 = 𝐵 ∧ 𝜓)
10		ceqsex2.3	. . 3 ⊢ 𝐴 ∈ V
11		ceqsex2.5	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
12	11	anbi2d 455	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑦 = 𝐵 ∧ 𝜑) ↔ (𝑦 = 𝐵 ∧ 𝜓)))
13	12	exbidv 1764	. . 3 ⊢ (𝑥 = 𝐴 → (∃𝑦(𝑦 = 𝐵 ∧ 𝜑) ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝜓)))
14	9, 10, 13	ceqsex 2679	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝜓))
15		ceqsex2.2	. . 3 ⊢ Ⅎ𝑦𝜒
16		ceqsex2.4	. . 3 ⊢ 𝐵 ∈ V
17		ceqsex2.6	. . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
18	15, 16, 17	ceqsex 2679	. 2 ⊢ (∃𝑦(𝑦 = 𝐵 ∧ 𝜓) ↔ 𝜒)
19	5, 14, 18	3bitri 205	1 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ 𝜒)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 930   = wceq 1299  Ⅎwnf 1404  ∃wex 1436   ∈ wcel 1448  Vcvv 2641

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-3an 932  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-v 2643

This theorem is referenced by:  ceqsex2v  2682