Theorem copsexg 5462

Description: Substitution of class 𝐴 for ordered pair ⟨𝑥, 𝑦⟩. Usage of this theorem is discouraged because it depends on ax-13 2405. Use the weaker copsexgw 5460 when possible. (Contributed by NM, 27-Dec-1996.) (Revised by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 25-Aug-2019.) (New usage is discouraged.)

Assertion

Ref	Expression
copsexg	⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem copsexg

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3460	. . . 4 ⊢ 𝑥 ∈ V
2		vex 3460	. . . 4 ⊢ 𝑦 ∈ V
3	1, 2	eqvinop 5457	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑧∃𝑤(𝐴 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩))
4		19.8a 2218	. . . . . . . . 9 ⊢ (∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5	4	19.23bi 2228	. . . . . . . 8 ⊢ ((⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
6	5	ex 416	. . . . . . 7 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ → (𝜑 → ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
7		vex 3460	. . . . . . . . 9 ⊢ 𝑧 ∈ V
8		vex 3460	. . . . . . . . 9 ⊢ 𝑤 ∈ V
9	7, 8	opth 5446	. . . . . . . 8 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑧 = 𝑥 ∧ 𝑤 = 𝑦))
10	9	anbi1i 633	. . . . . . . . . 10 ⊢ ((⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑))
11	10	2exbii 1871	. . . . . . . . 9 ⊢ (∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑))
12		nfe1 2186	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))
13		19.8a 2218	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 = 𝑦 ∧ 𝜑) → ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))
14	13	anim2i 626	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)) → (𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
15	14	anassrs 471	. . . . . . . . . . . . . 14 ⊢ (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) → (𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
16	15	eximi 1857	. . . . . . . . . . . . 13 ⊢ (∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) → ∃𝑦(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
17		biidd 264	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 𝑦 = 𝑥 → ((𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) ↔ (𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))))
18	17	drex1 2474	. . . . . . . . . . . . 13 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) ↔ ∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))))
19	16, 18	imbitrid 246	. . . . . . . . . . . 12 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) → ∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))))
20		anass 472	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ (𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)))
21	20	exbii 1870	. . . . . . . . . . . . . 14 ⊢ (∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ ∃𝑦(𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)))
22		19.40 1908	. . . . . . . . . . . . . . 15 ⊢ (∃𝑦(𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)) → (∃𝑦 𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
23		nfeqf2 2410	. . . . . . . . . . . . . . . . 17 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑧 = 𝑥)
24	23	19.9d 2240	. . . . . . . . . . . . . . . 16 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (∃𝑦 𝑧 = 𝑥 → 𝑧 = 𝑥))
25	24	anim1d 620	. . . . . . . . . . . . . . 15 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → ((∃𝑦 𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) → (𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))))
26	22, 25	syl5 34	. . . . . . . . . . . . . 14 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (∃𝑦(𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)) → (𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))))
27	21, 26	biimtrid 244	. . . . . . . . . . . . 13 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) → (𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))))
28		19.8a 2218	. . . . . . . . . . . . 13 ⊢ ((𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) → ∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
29	27, 28	syl6 35	. . . . . . . . . . . 12 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) → ∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))))
30	19, 29	pm2.61i 183	. . . . . . . . . . 11 ⊢ (∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) → ∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
31	12, 30	exlimi 2254	. . . . . . . . . 10 ⊢ (∃𝑥∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) → ∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
32		euequ 2626	. . . . . . . . . . . . . 14 ⊢ ∃!𝑥 𝑥 = 𝑧
33		equcom 2040	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
34	33	eubii 2614	. . . . . . . . . . . . . 14 ⊢ (∃!𝑥 𝑥 = 𝑧 ↔ ∃!𝑥 𝑧 = 𝑥)
35	32, 34	mpbi 232	. . . . . . . . . . . . 13 ⊢ ∃!𝑥 𝑧 = 𝑥
36		eupick 2662	. . . . . . . . . . . . 13 ⊢ ((∃!𝑥 𝑧 = 𝑥 ∧ ∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))) → (𝑧 = 𝑥 → ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
37	35, 36	mpan 700	. . . . . . . . . . . 12 ⊢ (∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) → (𝑧 = 𝑥 → ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
38	37	com12 32	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) → ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
39		euequ 2626	. . . . . . . . . . . . . 14 ⊢ ∃!𝑦 𝑦 = 𝑤
40		equcom 2040	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 ↔ 𝑤 = 𝑦)
41	40	eubii 2614	. . . . . . . . . . . . . 14 ⊢ (∃!𝑦 𝑦 = 𝑤 ↔ ∃!𝑦 𝑤 = 𝑦)
42	39, 41	mpbi 232	. . . . . . . . . . . . 13 ⊢ ∃!𝑦 𝑤 = 𝑦
43		eupick 2662	. . . . . . . . . . . . 13 ⊢ ((∃!𝑦 𝑤 = 𝑦 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) → (𝑤 = 𝑦 → 𝜑))
44	42, 43	mpan 700	. . . . . . . . . . . 12 ⊢ (∃𝑦(𝑤 = 𝑦 ∧ 𝜑) → (𝑤 = 𝑦 → 𝜑))
45	44	com12 32	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (∃𝑦(𝑤 = 𝑦 ∧ 𝜑) → 𝜑))
46	38, 45	sylan9 515	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) → 𝜑))
47	31, 46	syl5 34	. . . . . . . . 9 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (∃𝑥∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) → 𝜑))
48	11, 47	biimtrid 244	. . . . . . . 8 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝜑))
49	9, 48	sylbi 219	. . . . . . 7 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ → (∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝜑))
50	6, 49	impbid 214	. . . . . 6 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
51		eqeq1 2768	. . . . . . 7 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩))
52	51	anbi1d 640	. . . . . . . . 9 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
53	52	2exbidv 1946	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
54	53	bibi2d 344	. . . . . . 7 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → ((𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝜑 ↔ ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))))
55	51, 54	imbi12d 346	. . . . . 6 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → ((𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))) ↔ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))))
56	50, 55	mpbiri 260	. . . . 5 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))))
57	56	adantr 484	. . . 4 ⊢ ((𝐴 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))))
58	57	exlimivv 1954	. . 3 ⊢ (∃𝑧∃𝑤(𝐴 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))))
59	3, 58	sylbi 219	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))))
60	59	pm2.43i 52	1 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1560   = wceq 1562  ∃wex 1801  ∃!weu 2597  ⟨cop 4590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2214  ax-13 2405  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591

This theorem is referenced by:  opabid  5497