Theorem oteq1 4602

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 4593	. . 3 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
2	1	opeq1d 4599	. 2 ⊢ (𝐴 = 𝐵 → ⟨⟨𝐴, 𝐶⟩, 𝐷⟩ = ⟨⟨𝐵, 𝐶⟩, 𝐷⟩)
3		df-ot 4377	. 2 ⊢ ⟨𝐴, 𝐶, 𝐷⟩ = ⟨⟨𝐴, 𝐶⟩, 𝐷⟩
4		df-ot 4377	. 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 1653  ⟨cop 4374  ⟨cotp 4376

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-ot 4377

This theorem is referenced by:  oteq1d  4605  otiunsndisj  5176  efgi  18445  efgtf  18448  efgtval  18449  mapdh9a  37810  mapdh9aOLDN  37811  hdmapval2  37853  otiunsndisjX  42134