Theorem oteq1 4774

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 4763	. . 3 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
2	1	opeq1d 4771	. 2 ⊢ (𝐴 = 𝐵 → ⟨⟨𝐴, 𝐶⟩, 𝐷⟩ = ⟨⟨𝐵, 𝐶⟩, 𝐷⟩)
3		df-ot 4534	. 2 ⊢ ⟨𝐴, 𝐶, 𝐷⟩ = ⟨⟨𝐴, 𝐶⟩, 𝐷⟩
4		df-ot 4534	. 2 ⊢ ⟨𝐵, 𝐶, 𝐷⟩ = ⟨⟨𝐵, 𝐶⟩, 𝐷⟩
5	2, 3, 4	3eqtr4g 2858	1 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)

Syntax hints:   → wi 4   = wceq 1538  ⟨cop 4531  ⟨cotp 4533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-sn 4526  df-pr 4528  df-op 4532  df-ot 4534

This theorem is referenced by:  oteq1d  4777  otiunsndisj  5375  efgi  18837  efgtf  18840  efgtval  18841  mapdh9a  39085  mapdh9aOLDN  39086  hdmapval2  39128  otiunsndisjX  43835