Theorem oteq1 4875

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 4866	. . 3 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
2	1	opeq1d 4872	. 2 ⊢ (𝐴 = 𝐵 → ⟨⟨𝐴, 𝐶⟩, 𝐷⟩ = ⟨⟨𝐵, 𝐶⟩, 𝐷⟩)
3		df-ot 4630	. 2 ⊢ ⟨𝐴, 𝐶, 𝐷⟩ = ⟨⟨𝐴, 𝐶⟩, 𝐷⟩
4		df-ot 4630	. 2 ⊢ ⟨𝐵, 𝐶, 𝐷⟩ = ⟨⟨𝐵, 𝐶⟩, 𝐷⟩
5	2, 3, 4	3eqtr4g 2789	1 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)

Syntax hints:   → wi 4   = wceq 1533  ⟨cop 4627  ⟨cotp 4629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-ot 4630

This theorem is referenced by:  oteq1d  4878  otiunsndisj  5511  frxp3  8132  xpord3pred  8133  efgi  19635  efgtf  19638  efgtval  19639  mapdh9a  41163  mapdh9aOLDN  41164  hdmapval2  41206  otiunsndisjX  46532