Theorem opthg 1899

Description: Ordered pair theorem.

Hypotheses

Ref	Expression
opthg.1	⊢ A ∈ V
opthg.2	⊢ B ∈ V

Assertion

Ref	Expression
opthg	⊢ (D ∈ R → (⟨A, B⟩ = ⟨C, D⟩ ↔ (A = C ∧ B = D)))

Proof of Theorem opthg

Step	Hyp	Ref	Expression
1		opeq2 1877	. . 3 ⊢ (x = D → ⟨C, x⟩ = ⟨C, D⟩)
2	1	cleq2d 1112	. 2 ⊢ (x = D → (⟨A, B⟩ = ⟨C, x⟩ ↔ ⟨A, B⟩ = ⟨C, D⟩))
3		cleq2 1110	. . 3 ⊢ (x = D → (B = x ↔ B = D))
4	3	anbi2d 468	. 2 ⊢ (x = D → ((A = C ∧ B = x) ↔ (A = C ∧ B = D)))
5		opthg.1	. . 3 ⊢ A ∈ V
6		opthg.2	. . 3 ⊢ B ∈ V
7		visset 1350	. . 3 ⊢ x ∈ V
8	5, 6, 7	opth 1898	. 2 ⊢ (⟨A, B⟩ = ⟨C, x⟩ ↔ (A = C ∧ B = x))
9	2, 4, 8	vtoclbg 1384	1 ⊢ (D ∈ R → (⟨A, B⟩ = ⟨C, D⟩ ↔ (A = C ∧ B = D)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810

This theorem is referenced by:  otthg 1900  copsex4g 1904  dmsnop 2547

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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