Theorem oteq1 4888

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 4879	. . 3 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
2	1	opeq1d 4885	. 2 ⊢ (𝐴 = 𝐵 → ⟨⟨𝐴, 𝐶⟩, 𝐷⟩ = ⟨⟨𝐵, 𝐶⟩, 𝐷⟩)
3		df-ot 4642	. 2 ⊢ ⟨𝐴, 𝐶, 𝐷⟩ = ⟨⟨𝐴, 𝐶⟩, 𝐷⟩
4		df-ot 4642	. 2 ⊢ ⟨𝐵, 𝐶, 𝐷⟩ = ⟨⟨𝐵, 𝐶⟩, 𝐷⟩
5	2, 3, 4	3eqtr4g 2791	1 ⊢ (𝐴 = 𝐵 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)

Syntax hints:   → wi 4   = wceq 1534  ⟨cop 4639  ⟨cotp 4641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-ot 4642

This theorem is referenced by:  oteq1d  4891  otiunsndisj  5526  frxp3  8165  xpord3pred  8166  efgi  19717  efgtf  19720  efgtval  19721  mapdh9a  41488  mapdh9aOLDN  41489  hdmapval2  41531  otiunsndisjX  46892