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Theorem opthg2 3967
Description: Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opthg2 ((𝐶 𝑉 𝐷 𝑊) → (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ ↔ (A = 𝐶 B = 𝐷)))

Proof of Theorem opthg2
StepHypRef Expression
1 opthg 3966 . 2 ((𝐶 𝑉 𝐷 𝑊) → (⟨𝐶, 𝐷⟩ = ⟨A, B⟩ ↔ (𝐶 = A 𝐷 = B)))
2 eqcom 2039 . 2 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨A, B⟩)
3 eqcom 2039 . . 3 (A = 𝐶𝐶 = A)
4 eqcom 2039 . . 3 (B = 𝐷𝐷 = B)
53, 4anbi12i 433 . 2 ((A = 𝐶 B = 𝐷) ↔ (𝐶 = A 𝐷 = B))
61, 2, 53bitr4g 212 1 ((𝐶 𝑉 𝐷 𝑊) → (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ ↔ (A = 𝐶 B = 𝐷)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  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-bnd 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:  opth2  3968  fliftel  5376  axprecex  6744
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