Theorem oteq3 3724

Description: Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)

Assertion

Ref	Expression
oteq3	⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)

Proof of Theorem oteq3

Step	Hyp	Ref	Expression
1		opeq2 3714	. 2 ⊢ (𝐴 = 𝐵 → ⟨⟨𝐶, 𝐷⟩, 𝐴⟩ = ⟨⟨𝐶, 𝐷⟩, 𝐵⟩)
2		df-ot 3542	. 2 ⊢ ⟨𝐶, 𝐷, 𝐴⟩ = ⟨⟨𝐶, 𝐷⟩, 𝐴⟩
3		df-ot 3542	. 2 ⊢ ⟨𝐶, 𝐷, 𝐵⟩ = ⟨⟨𝐶, 𝐷⟩, 𝐵⟩
4	1, 2, 3	3eqtr4g 2198	1 ⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)

Syntax hints:   → wi 4   = wceq 1332  ⟨cop 3535  ⟨cotp 3536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-ot 3542

This theorem is referenced by:  oteq3d  3727