Theorem oteqex 5355

Description: Equivalence of existence implied by equality of ordered triples. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
oteqex	⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ↔ (𝑅 ∈ V ∧ 𝑆 ∈ V ∧ 𝑇 ∈ V)))

Proof of Theorem oteqex

Step	Hyp	Ref	Expression
1		simp3 1135	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐶 ∈ V)
2	1	a1i 11	. 2 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐶 ∈ V))
3		simp3 1135	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V ∧ 𝑇 ∈ V) → 𝑇 ∈ V)
4		oteqex2 5354	. . 3 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ → (𝐶 ∈ V ↔ 𝑇 ∈ V))
5	3, 4	syl5ibr 249	. 2 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ → ((𝑅 ∈ V ∧ 𝑆 ∈ V ∧ 𝑇 ∈ V) → 𝐶 ∈ V))
6		opex 5321	. . . . . . . 8 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
7		opthg 5334	. . . . . . . 8 ⊢ ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ V) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑅, 𝑆⟩ ∧ 𝐶 = 𝑇)))
8	6, 7	mpan 689	. . . . . . 7 ⊢ (𝐶 ∈ V → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑅, 𝑆⟩ ∧ 𝐶 = 𝑇)))
9	8	simprbda 502	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩) → ⟨𝐴, 𝐵⟩ = ⟨𝑅, 𝑆⟩)
10		opeqex 5353	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑅, 𝑆⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑅 ∈ V ∧ 𝑆 ∈ V)))
11	9, 10	syl 17	. . . . 5 ⊢ ((𝐶 ∈ V ∧ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝑅 ∈ V ∧ 𝑆 ∈ V)))
12	4	adantl 485	. . . . 5 ⊢ ((𝐶 ∈ V ∧ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩) → (𝐶 ∈ V ↔ 𝑇 ∈ V))
13	11, 12	anbi12d 633	. . . 4 ⊢ ((𝐶 ∈ V ∧ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩) → (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐶 ∈ V) ↔ ((𝑅 ∈ V ∧ 𝑆 ∈ V) ∧ 𝑇 ∈ V)))
14		df-3an 1086	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐶 ∈ V))
15		df-3an 1086	. . . 4 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V ∧ 𝑇 ∈ V) ↔ ((𝑅 ∈ V ∧ 𝑆 ∈ V) ∧ 𝑇 ∈ V))
16	13, 14, 15	3bitr4g 317	. . 3 ⊢ ((𝐶 ∈ V ∧ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩) → ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ↔ (𝑅 ∈ V ∧ 𝑆 ∈ V ∧ 𝑇 ∈ V)))
17	16	expcom 417	. 2 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ → (𝐶 ∈ V → ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ↔ (𝑅 ∈ V ∧ 𝑆 ∈ V ∧ 𝑇 ∈ V))))
18	2, 5, 17	pm5.21ndd 384	1 ⊢ (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ = ⟨⟨𝑅, 𝑆⟩, 𝑇⟩ → ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ↔ (𝑅 ∈ V ∧ 𝑆 ∈ V ∧ 𝑇 ∈ V)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ⟨cop 4531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532

This theorem is referenced by: (None)