Theorem copsex2t 4162

Description: Closed theorem form of copsex2g 4163. (Contributed by NM, 17-Feb-2013.)

Assertion

Ref	Expression
copsex2t	⊢ ((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))

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

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem copsex2t

Step	Hyp	Ref	Expression
1		elisset 2695	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		elisset 2695	. . . 4 ⊢ (𝐵 ∈ 𝑊 → ∃𝑦 𝑦 = 𝐵)
3	1, 2	anim12i 336	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
4		eeanv 1902	. . 3 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
5	3, 4	sylibr 133	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
6		nfa1 1521	. . . 4 ⊢ Ⅎ𝑥∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
7		nfe1 1472	. . . . 5 ⊢ Ⅎ𝑥∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
8		nfv 1508	. . . . 5 ⊢ Ⅎ𝑥𝜓
9	7, 8	nfbi 1568	. . . 4 ⊢ Ⅎ𝑥(∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓)
10		nfa2 1558	. . . . 5 ⊢ Ⅎ𝑦∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
11		nfe1 1472	. . . . . . 7 ⊢ Ⅎ𝑦∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
12	11	nfex 1616	. . . . . 6 ⊢ Ⅎ𝑦∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
13		nfv 1508	. . . . . 6 ⊢ Ⅎ𝑦𝜓
14	12, 13	nfbi 1568	. . . . 5 ⊢ Ⅎ𝑦(∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓)
15		opeq12 3702	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
16		copsexg 4161	. . . . . . . . . 10 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
17	16	eqcoms 2140	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → (𝜑 ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
18	15, 17	syl 14	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
19	18	adantl 275	. . . . . . 7 ⊢ ((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜑 ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
20		sp 1488	. . . . . . . . 9 ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) → ∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)))
21	20	19.21bi 1537	. . . . . . . 8 ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)))
22	21	imp 123	. . . . . . 7 ⊢ ((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜑 ↔ 𝜓))
23	19, 22	bitr3d 189	. . . . . 6 ⊢ ((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
24	23	ex 114	. . . . 5 ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓)))
25	10, 14, 24	exlimd 1576	. . . 4 ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) → (∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓)))
26	6, 9, 25	exlimd 1576	. . 3 ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) → (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓)))
27	26	imp 123	. 2 ⊢ ((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ∧ ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
28	5, 27	sylan2 284	1 ⊢ ((∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ⟨cop 3525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531

This theorem is referenced by:  opelopabt  4179