Theorem copsex4g 1904

Description: An implicit substitution inference for 2 ordered pairs.

Hypothesis

Ref	Expression
copsex4g.1	⊢ (((x = A ∧ y = B) ∧ (z = C ∧ w = D)) → (φ ↔ ψ))

Assertion

Ref	Expression
copsex4g	⊢ (((A ∈ R ∧ B ∈ S) ∧ (C ∈ R ∧ D ∈ S)) → (∃x∃y∃z∃w((⟨A, B⟩ = ⟨x, y⟩ ∧ ⟨C, D⟩ = ⟨z, w⟩) ∧ φ) ↔ ψ))

Distinct variable group(s):   x,y,z,w,A   x,B,y,z,w   x,C,y,z,w   x,D,y,z,w   ψ,x,y,z,w   x,R,y,z,w   x,S,y,z,w

Proof of Theorem copsex4g

Step	Hyp	Ref	Expression
1		visset 1350	. . . . . . . 8 ⊢ x ∈ V
2		visset 1350	. . . . . . . 8 ⊢ y ∈ V
3	1, 2	opthg 1899	. . . . . . 7 ⊢ (B ∈ S → (⟨x, y⟩ = ⟨A, B⟩ ↔ (x = A ∧ y = B)))
4		cleqcom 1103	. . . . . . 7 ⊢ (⟨A, B⟩ = ⟨x, y⟩ ↔ ⟨x, y⟩ = ⟨A, B⟩)
5	3, 4	syl5bb 410	. . . . . 6 ⊢ (B ∈ S → (⟨A, B⟩ = ⟨x, y⟩ ↔ (x = A ∧ y = B)))
6	5	adantl 305	. . . . 5 ⊢ ((A ∈ R ∧ B ∈ S) → (⟨A, B⟩ = ⟨x, y⟩ ↔ (x = A ∧ y = B)))
7		visset 1350	. . . . . . . 8 ⊢ z ∈ V
8		visset 1350	. . . . . . . 8 ⊢ w ∈ V
9	7, 8	opthg 1899	. . . . . . 7 ⊢ (D ∈ S → (⟨z, w⟩ = ⟨C, D⟩ ↔ (z = C ∧ w = D)))
10		cleqcom 1103	. . . . . . 7 ⊢ (⟨C, D⟩ = ⟨z, w⟩ ↔ ⟨z, w⟩ = ⟨C, D⟩)
11	9, 10	syl5bb 410	. . . . . 6 ⊢ (D ∈ S → (⟨C, D⟩ = ⟨z, w⟩ ↔ (z = C ∧ w = D)))
12	11	adantl 305	. . . . 5 ⊢ ((C ∈ R ∧ D ∈ S) → (⟨C, D⟩ = ⟨z, w⟩ ↔ (z = C ∧ w = D)))
13	6, 12	bi2anan9 478	. . . 4 ⊢ (((A ∈ R ∧ B ∈ S) ∧ (C ∈ R ∧ D ∈ S)) → ((⟨A, B⟩ = ⟨x, y⟩ ∧ ⟨C, D⟩ = ⟨z, w⟩) ↔ ((x = A ∧ y = B) ∧ (z = C ∧ w = D))))
14	13	anbi1d 469	. . 3 ⊢ (((A ∈ R ∧ B ∈ S) ∧ (C ∈ R ∧ D ∈ S)) → (((⟨A, B⟩ = ⟨x, y⟩ ∧ ⟨C, D⟩ = ⟨z, w⟩) ∧ φ) ↔ (((x = A ∧ y = B) ∧ (z = C ∧ w = D)) ∧ φ)))
15	14	bi4exdv 940	. 2 ⊢ (((A ∈ R ∧ B ∈ S) ∧ (C ∈ R ∧ D ∈ S)) → (∃x∃y∃z∃w((⟨A, B⟩ = ⟨x, y⟩ ∧ ⟨C, D⟩ = ⟨z, w⟩) ∧ φ) ↔ ∃x∃y∃z∃w(((x = A ∧ y = B) ∧ (z = C ∧ w = D)) ∧ φ)))
16		id 9	. . 3 ⊢ (((x = A ∧ y = B) ∧ (z = C ∧ w = D)) → ((x = A ∧ y = B) ∧ (z = C ∧ w = D)))
17		copsex4g.1	. . 3 ⊢ (((x = A ∧ y = B) ∧ (z = C ∧ w = D)) → (φ ↔ ψ))
18	16, 17	cgsex4g 1369	. 2 ⊢ (((A ∈ R ∧ B ∈ S) ∧ (C ∈ R ∧ D ∈ S)) → (∃x∃y∃z∃w(((x = A ∧ y = B) ∧ (z = C ∧ w = D)) ∧ φ) ↔ ψ))
19	15, 18	bitrd 406	1 ⊢ (((A ∈ R ∧ B ∈ S) ∧ (C ∈ R ∧ D ∈ S)) → (∃x∃y∃z∃w((⟨A, B⟩ = ⟨x, y⟩ ∧ ⟨C, D⟩ = ⟨z, w⟩) ∧ φ) ↔ ψ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ⟨cop 1810

This theorem is referenced by:  opbrop 2472  oprabval3 3052

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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