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Theorem opthg2 3946
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 3945 . 2 ((𝐶 𝑉 𝐷 𝑊) → (⟨𝐶, 𝐷⟩ = ⟨A, B⟩ ↔ (𝐶 = A 𝐷 = B)))
2 eqcom 2020 . 2 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨A, B⟩)
3 eqcom 2020 . . 3 (A = 𝐶𝐶 = A)
4 eqcom 2020 . . 3 (B = 𝐷𝐷 = B)
53, 4anbi12i 436 . 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 1226   wcel 1370  cop 3349
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355
This theorem is referenced by:  opth2  3947  fliftel  5354  axprecex  6568
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