Theorem oteq1 4863

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

Assertion

Ref	Expression
oteq1	⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)

Proof of Theorem oteq1

Step	Hyp	Ref	Expression
1		opeq1 4843	. . 3 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
2	1	opeq1d 4849	. 2 ⊢ (𝐴 = 𝐵 → ⟨⟨𝐴, 𝐶⟩, 𝐷⟩ = ⟨⟨𝐵, 𝐶⟩, 𝐷⟩)
3		df-ot 4862	. 2 ⊢ ⟨𝐴, 𝐶, 𝐷⟩ = ⟨⟨𝐴, 𝐶⟩, 𝐷⟩
4		df-ot 4862	. 2 ⊢ ⟨𝐵, 𝐶, 𝐷⟩ = ⟨⟨𝐵, 𝐶⟩, 𝐷⟩
5	2, 3, 4	3eqtr4g 2823	1 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ⟨cop 4838  ⟨cotp 4861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-ot 4862

This theorem is used by:  oteq1d  4866  eqvinot  5461  otiunsndisj  5497  frxp3  8195  xpord3pred  8196  efgi  19963  efgtf  19966  efgtval  19967  dfk12  36012  dfk22  36013  dfk32  36014  mapdh9a  43038  mapdh9aOLDN  43039  hdmapval2  43081  otiunsndisjX  48554