Theorem oteq3 3769

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 3759	. 2 ⊢ (𝐴 = 𝐵 → ⟨⟨𝐶, 𝐷⟩, 𝐴⟩ = ⟨⟨𝐶, 𝐷⟩, 𝐵⟩)
2		df-ot 3586	. 2 ⊢ ⟨𝐶, 𝐷, 𝐴⟩ = ⟨⟨𝐶, 𝐷⟩, 𝐴⟩
3		df-ot 3586	. 2 ⊢ ⟨𝐶, 𝐷, 𝐵⟩ = ⟨⟨𝐶, 𝐷⟩, 𝐵⟩
4	1, 2, 3	3eqtr4g 2224	1 ⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)

Syntax hints:   → wi 4   = wceq 1343  ⟨cop 3579  ⟨cotp 3580

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-ot 3586

This theorem is referenced by:  oteq3d  3772