Theorem oteq3 4776

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

Assertion

Ref	Expression
oteq3	⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)

Proof of Theorem oteq3

Step	Hyp	Ref	Expression
1		opeq2 4765	. 2 ⊢ (𝐴 = 𝐵 → ⟨⟨𝐶, 𝐷⟩, 𝐴⟩ = ⟨⟨𝐶, 𝐷⟩, 𝐵⟩)
2		df-ot 4534	. 2 ⊢ ⟨𝐶, 𝐷, 𝐴⟩ = ⟨⟨𝐶, 𝐷⟩, 𝐴⟩
3		df-ot 4534	. 2 ⊢ ⟨𝐶, 𝐷, 𝐵⟩ = ⟨⟨𝐶, 𝐷⟩, 𝐵⟩
4	1, 2, 3	3eqtr4g 2858	1 ⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)

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:  oteq3d  4779  otsndisj  5374  otiunsndisj  5375  efgi0  18838  efgi1  18839  mapdhcl  39023  mapdh6dN  39035  mapdh8  39084  mapdh9a  39085  mapdh9aOLDN  39086  hdmap1l6d  39109  hdmapval  39124  hdmapval2  39128  hdmapval3N  39134  otiunsndisjX  43835