Theorem otthg 1900

Description: Ordered triple theorem.

Hypotheses

Ref	Expression
otthg.1	⊢ A ∈ V
otthg.2	⊢ B ∈ V
otthg.3	⊢ R ∈ V

Assertion

Ref	Expression
otthg	⊢ ((D ∈ F ∧ S ∈ G) → (⟨⟨A, B⟩, R⟩ = ⟨⟨C, D⟩, S⟩ ↔ (A = C ∧ B = D ∧ R = S)))

Proof of Theorem otthg

Step	Hyp	Ref	Expression
1		opex 1893	. . . 4 ⊢ ⟨A, B⟩ ∈ V
2		otthg.3	. . . 4 ⊢ R ∈ V
3	1, 2	opthg 1899	. . 3 ⊢ (S ∈ G → (⟨⟨A, B⟩, R⟩ = ⟨⟨C, D⟩, S⟩ ↔ (⟨A, B⟩ = ⟨C, D⟩ ∧ R = S)))
4		otthg.1	. . . . 5 ⊢ A ∈ V
5		otthg.2	. . . . 5 ⊢ B ∈ V
6	4, 5	opthg 1899	. . . 4 ⊢ (D ∈ F → (⟨A, B⟩ = ⟨C, D⟩ ↔ (A = C ∧ B = D)))
7	6	anbi1d 469	. . 3 ⊢ (D ∈ F → ((⟨A, B⟩ = ⟨C, D⟩ ∧ R = S) ↔ ((A = C ∧ B = D) ∧ R = S)))
8	3, 7	sylan9bbr 419	. 2 ⊢ ((D ∈ F ∧ S ∈ G) → (⟨⟨A, B⟩, R⟩ = ⟨⟨C, D⟩, S⟩ ↔ ((A = C ∧ B = D) ∧ R = S)))
9		df-3an 583	. 2 ⊢ ((A = C ∧ B = D ∧ R = S) ↔ ((A = C ∧ B = D) ∧ R = S))
10	8, 9	syl6bbr 416	1 ⊢ ((D ∈ F ∧ S ∈ G) → (⟨⟨A, B⟩, R⟩ = ⟨⟨C, D⟩, S⟩ ↔ (A = C ∧ B = D ∧ R = S)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   ∧ w3a 581   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810

This theorem is referenced by:  eloprabg 3035

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-3an 583  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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