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Theorem otth2 3969
Description: Ordered triple theorem, with triple express with ordered pairs. (Contributed by NM, 1-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
otth.1 A V
otth.2 B V
otth.3 𝑅 V
Assertion
Ref Expression
otth2 (⟨⟨A, B⟩, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩ ↔ (A = 𝐶 B = 𝐷 𝑅 = 𝑆))

Proof of Theorem otth2
StepHypRef Expression
1 otth.1 . . . 4 A V
2 otth.2 . . . 4 B V
31, 2opth 3965 . . 3 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ ↔ (A = 𝐶 B = 𝐷))
43anbi1i 431 . 2 ((⟨A, B⟩ = ⟨𝐶, 𝐷 𝑅 = 𝑆) ↔ ((A = 𝐶 B = 𝐷) 𝑅 = 𝑆))
5 opexgOLD 3956 . . . 4 ((A V B V) → ⟨A, B V)
61, 2, 5mp2an 402 . . 3 A, B V
7 otth.3 . . 3 𝑅 V
86, 7opth 3965 . 2 (⟨⟨A, B⟩, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩ ↔ (⟨A, B⟩ = ⟨𝐶, 𝐷 𝑅 = 𝑆))
9 df-3an 886 . 2 ((A = 𝐶 B = 𝐷 𝑅 = 𝑆) ↔ ((A = 𝐶 B = 𝐷) 𝑅 = 𝑆))
104, 8, 93bitr4i 201 1 (⟨⟨A, B⟩, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩ ↔ (A = 𝐶 B = 𝐷 𝑅 = 𝑆))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  Vcvv 2551  cop 3370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376
This theorem is referenced by:  otth  3970  oprabid  5480  eloprabga  5533
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