Theorem ceqsex2 2639

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 923	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
2	1	exbii 1536	. . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ ∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
3		19.42v 1827	. . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ (𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)))
4	2, 3	bitri 182	. . 3 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)))
5	4	exbii 1536	. 2 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)))
6		nfv 1461	. . . . 5 ⊢ Ⅎ𝑥 𝑦 = 𝐵
7		ceqsex2.1	. . . . 5 ⊢ Ⅎ𝑥𝜓
8	6, 7	nfan 1497	. . . 4 ⊢ Ⅎ𝑥(𝑦 = 𝐵 ∧ 𝜓)
9	8	nfex 1568	. . 3 ⊢ Ⅎ𝑥∃𝑦(𝑦 = 𝐵 ∧ 𝜓)
10		ceqsex2.3	. . 3 ⊢ 𝐴 ∈ V
11		ceqsex2.5	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
12	11	anbi2d 451	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑦 = 𝐵 ∧ 𝜑) ↔ (𝑦 = 𝐵 ∧ 𝜓)))
13	12	exbidv 1746	. . 3 ⊢ (𝑥 = 𝐴 → (∃𝑦(𝑦 = 𝐵 ∧ 𝜑) ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝜓)))
14	9, 10, 13	ceqsex 2637	. 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 2637	. 2 ⊢ (∃𝑦(𝑦 = 𝐵 ∧ 𝜓) ↔ 𝜒)
19	5, 14, 18	3bitri 204	1 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ 𝜒)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∧ w3a 919   = wceq 1284  Ⅎwnf 1389  ∃wex 1421   ∈ wcel 1433  Vcvv 2601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-v 2603

This theorem is referenced by:  ceqsex2v  2640