Theorem copsexg 1902

Description: Substitution of class A for ordered pair ⟨x, y⟩.

Assertion

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

Distinct variable group(s):   x,y,A

Proof of Theorem copsexg

Step	Hyp	Ref	Expression
1		visset 1350	. . . 4 ⊢ x ∈ V
2		visset 1350	. . . 4 ⊢ y ∈ V
3	1, 2	eqvinop 1901	. . 3 ⊢ (A = ⟨x, y⟩ ↔ ∃z∃w(A = ⟨z, w⟩ ∧ ⟨z, w⟩ = ⟨x, y⟩))
4		cleqcom 1103	. . . . . . . . 9 ⊢ (⟨z, w⟩ = ⟨x, y⟩ ↔ ⟨x, y⟩ = ⟨z, w⟩)
5		visset 1350	. . . . . . . . . 10 ⊢ w ∈ V
6	1, 2, 5	opth 1898	. . . . . . . . 9 ⊢ (⟨x, y⟩ = ⟨z, w⟩ ↔ (x = z ∧ y = w))
7	4, 6	bitr 151	. . . . . . . 8 ⊢ (⟨z, w⟩ = ⟨x, y⟩ ↔ (x = z ∧ y = w))
8		ceqex 1410	. . . . . . . . 9 ⊢ (y = w → (φ ↔ ∃y(y = w ∧ φ)))
9		ceqex 1410	. . . . . . . . 9 ⊢ (x = z → (∃y(y = w ∧ φ) ↔ ∃x(x = z ∧ ∃y(y = w ∧ φ))))
10	8, 9	sylan9bbr 419	. . . . . . . 8 ⊢ ((x = z ∧ y = w) → (φ ↔ ∃x(x = z ∧ ∃y(y = w ∧ φ))))
11	7, 10	sylbi 174	. . . . . . 7 ⊢ (⟨z, w⟩ = ⟨x, y⟩ → (φ ↔ ∃x(x = z ∧ ∃y(y = w ∧ φ))))
12	7	anbi1i 368	. . . . . . . . . . 11 ⊢ ((⟨z, w⟩ = ⟨x, y⟩ ∧ φ) ↔ ((x = z ∧ y = w) ∧ φ))
13		anass 336	. . . . . . . . . . 11 ⊢ (((x = z ∧ y = w) ∧ φ) ↔ (x = z ∧ (y = w ∧ φ)))
14	12, 13	bitr 151	. . . . . . . . . 10 ⊢ ((⟨z, w⟩ = ⟨x, y⟩ ∧ φ) ↔ (x = z ∧ (y = w ∧ φ)))
15	14	biex 733	. . . . . . . . 9 ⊢ (∃y(⟨z, w⟩ = ⟨x, y⟩ ∧ φ) ↔ ∃y(x = z ∧ (y = w ∧ φ)))
16		19.42v 966	. . . . . . . . 9 ⊢ (∃y(x = z ∧ (y = w ∧ φ)) ↔ (x = z ∧ ∃y(y = w ∧ φ)))
17	15, 16	bitr 151	. . . . . . . 8 ⊢ (∃y(⟨z, w⟩ = ⟨x, y⟩ ∧ φ) ↔ (x = z ∧ ∃y(y = w ∧ φ)))
18	17	biex 733	. . . . . . 7 ⊢ (∃x∃y(⟨z, w⟩ = ⟨x, y⟩ ∧ φ) ↔ ∃x(x = z ∧ ∃y(y = w ∧ φ)))
19	11, 18	syl6bbr 416	. . . . . 6 ⊢ (⟨z, w⟩ = ⟨x, y⟩ → (φ ↔ ∃x∃y(⟨z, w⟩ = ⟨x, y⟩ ∧ φ)))
20		cleq1 1107	. . . . . . 7 ⊢ (A = ⟨z, w⟩ → (A = ⟨x, y⟩ ↔ ⟨z, w⟩ = ⟨x, y⟩))
21	20	anbi1d 469	. . . . . . . . 9 ⊢ (A = ⟨z, w⟩ → ((A = ⟨x, y⟩ ∧ φ) ↔ (⟨z, w⟩ = ⟨x, y⟩ ∧ φ)))
22	21	bi2exdv 938	. . . . . . . 8 ⊢ (A = ⟨z, w⟩ → (∃x∃y(A = ⟨x, y⟩ ∧ φ) ↔ ∃x∃y(⟨z, w⟩ = ⟨x, y⟩ ∧ φ)))
23	22	bibi2d 470	. . . . . . 7 ⊢ (A = ⟨z, w⟩ → ((φ ↔ ∃x∃y(A = ⟨x, y⟩ ∧ φ)) ↔ (φ ↔ ∃x∃y(⟨z, w⟩ = ⟨x, y⟩ ∧ φ))))
24	20, 23	imbi12d 474	. . . . . 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)))))
25	19, 24	mpbiri 169	. . . . 5 ⊢ (A = ⟨z, w⟩ → (A = ⟨x, y⟩ → (φ ↔ ∃x∃y(A = ⟨x, y⟩ ∧ φ))))
26	25	adantr 306	. . . 4 ⊢ ((A = ⟨z, w⟩ ∧ ⟨z, w⟩ = ⟨x, y⟩) → (A = ⟨x, y⟩ → (φ ↔ ∃x∃y(A = ⟨x, y⟩ ∧ φ))))
27	26	19.23aivv 953	. . 3 ⊢ (∃z∃w(A = ⟨z, w⟩ ∧ ⟨z, w⟩ = ⟨x, y⟩) → (A = ⟨x, y⟩ → (φ ↔ ∃x∃y(A = ⟨x, y⟩ ∧ φ))))
28	3, 27	sylbi 174	. 2 ⊢ (A = ⟨x, y⟩ → (A = ⟨x, y⟩ → (φ ↔ ∃x∃y(A = ⟨x, y⟩ ∧ φ))))
29	28	pm2.43i 58	1 ⊢ (A = ⟨x, y⟩ → (φ ↔ ∃x∃y(A = ⟨x, y⟩ ∧ φ)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = weq 797   = wceq 1091  ⟨cop 1810

This theorem is referenced by:  copsex2g 1903  mosubop 1911  ssopab2 2119

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815

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