Theorem copsexg 5187

Description: Substitution of class 𝐴 for ordered pair ⟨𝑥, 𝑦⟩. (Contributed by NM, 27-Dec-1996.) (Revised by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 25-Aug-2019.)

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 3401	. . . 4 ⊢ 𝑥 ∈ V
2		vex 3401	. . . 4 ⊢ 𝑦 ∈ V
3	1, 2	eqvinop 5186	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑧∃𝑤(𝐴 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩))
4		19.8a 2166	. . . . . . . . 9 ⊢ (∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5	4	19.23bi 2175	. . . . . . . 8 ⊢ ((⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
6	5	ex 403	. . . . . . 7 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ → (𝜑 → ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
7		vex 3401	. . . . . . . . 9 ⊢ 𝑧 ∈ V
8		vex 3401	. . . . . . . . 9 ⊢ 𝑤 ∈ V
9	7, 8	opth 5176	. . . . . . . 8 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑧 = 𝑥 ∧ 𝑤 = 𝑦))
10	9	anbi1i 617	. . . . . . . . . 10 ⊢ ((⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑))
11	10	2exbii 1893	. . . . . . . . 9 ⊢ (∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑))
12		nfe1 2144	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))
13		19.8a 2166	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 = 𝑦 ∧ 𝜑) → ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))
14	13	anim2i 610	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)) → (𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
15	14	anassrs 461	. . . . . . . . . . . . . 14 ⊢ (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) → (𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
16	15	eximi 1878	. . . . . . . . . . . . 13 ⊢ (∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) → ∃𝑦(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
17		biidd 254	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 𝑦 = 𝑥 → ((𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) ↔ (𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))))
18	17	drex1 2407	. . . . . . . . . . . . 13 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) ↔ ∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))))
19	16, 18	syl5ib 236	. . . . . . . . . . . 12 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) → ∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))))
20		anass 462	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ (𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)))
21	20	exbii 1892	. . . . . . . . . . . . . 14 ⊢ (∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ ∃𝑦(𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)))
22		19.40 1915	. . . . . . . . . . . . . . 15 ⊢ (∃𝑦(𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)) → (∃𝑦 𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
23		nfeqf2 2339	. . . . . . . . . . . . . . . . 17 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑧 = 𝑥)
24	23	19.9d 2187	. . . . . . . . . . . . . . . 16 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (∃𝑦 𝑧 = 𝑥 → 𝑧 = 𝑥))
25	24	anim1d 604	. . . . . . . . . . . . . . 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	syl5bi 234	. . . . . . . . . . . . 13 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) → (𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))))
28		19.8a 2166	. . . . . . . . . . . . 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 177	. . . . . . . . . . 11 ⊢ (∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) → ∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
31	12, 30	exlimi 2203	. . . . . . . . . 10 ⊢ (∃𝑥∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) → ∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
32		euequ 2616	. . . . . . . . . . . . . 14 ⊢ ∃!𝑥 𝑥 = 𝑧
33		equcom 2065	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
34	33	eubii 2605	. . . . . . . . . . . . . 14 ⊢ (∃!𝑥 𝑥 = 𝑧 ↔ ∃!𝑥 𝑧 = 𝑥)
35	32, 34	mpbi 222	. . . . . . . . . . . . 13 ⊢ ∃!𝑥 𝑧 = 𝑥
36		eupick 2663	. . . . . . . . . . . . 13 ⊢ ((∃!𝑥 𝑧 = 𝑥 ∧ ∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))) → (𝑧 = 𝑥 → ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
37	35, 36	mpan 680	. . . . . . . . . . . 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 2616	. . . . . . . . . . . . . 14 ⊢ ∃!𝑦 𝑦 = 𝑤
40		equcom 2065	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 ↔ 𝑤 = 𝑦)
41	40	eubii 2605	. . . . . . . . . . . . . 14 ⊢ (∃!𝑦 𝑦 = 𝑤 ↔ ∃!𝑦 𝑤 = 𝑦)
42	39, 41	mpbi 222	. . . . . . . . . . . . 13 ⊢ ∃!𝑦 𝑤 = 𝑦
43		eupick 2663	. . . . . . . . . . . . 13 ⊢ ((∃!𝑦 𝑤 = 𝑦 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) → (𝑤 = 𝑦 → 𝜑))
44	42, 43	mpan 680	. . . . . . . . . . . 12 ⊢ (∃𝑦(𝑤 = 𝑦 ∧ 𝜑) → (𝑤 = 𝑦 → 𝜑))
45	44	com12 32	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (∃𝑦(𝑤 = 𝑦 ∧ 𝜑) → 𝜑))
46	38, 45	sylan9 503	. . . . . . . . . 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	syl5bi 234	. . . . . . . 8 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝜑))
49	9, 48	sylbi 209	. . . . . . 7 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ → (∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝜑))
50	6, 49	impbid 204	. . . . . 6 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
51		eqeq1 2782	. . . . . . 7 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩))
52	51	anbi1d 623	. . . . . . . . 9 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
53	52	2exbidv 1967	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
54	53	bibi2d 334	. . . . . . 7 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → ((𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝜑 ↔ ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))))
55	51, 54	imbi12d 336	. . . . . 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 250	. . . . 5 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))))
57	56	adantr 474	. . . 4 ⊢ ((𝐴 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))))
58	57	exlimivv 1975	. . 3 ⊢ (∃𝑧∃𝑤(𝐴 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))))
59	3, 58	sylbi 209	. 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 198   ∧ wa 386  ∀wal 1599   = wceq 1601  ∃wex 1823  ∃!weu 2586  ⟨cop 4404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405

This theorem is referenced by:  copsex2t  5188  copsex2g  5189  mosubopt  5207  opabid  5219  brabgaf  29983