Theorem oteq3 4812

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 4802	. 2 ⊢ (𝐴 = 𝐵 → ⟨⟨𝐶, 𝐷⟩, 𝐴⟩ = ⟨⟨𝐶, 𝐷⟩, 𝐵⟩)
2		df-ot 4567	. 2 ⊢ ⟨𝐶, 𝐷, 𝐴⟩ = ⟨⟨𝐶, 𝐷⟩, 𝐴⟩
3		df-ot 4567	. 2 ⊢ ⟨𝐶, 𝐷, 𝐵⟩ = ⟨⟨𝐶, 𝐷⟩, 𝐵⟩
4	1, 2, 3	3eqtr4g 2804	1 ⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)

Syntax hints:   → wi 4   = wceq 1539  ⟨cop 4564  ⟨cotp 4566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-ot 4567

This theorem is referenced by:  oteq3d  4815  otsndisj  5427  otiunsndisj  5428  efgi0  19241  efgi1  19242  mapdhcl  39668  mapdh6dN  39680  mapdh8  39729  mapdh9a  39730  mapdh9aOLDN  39731  hdmap1l6d  39754  hdmapval  39769  hdmapval2  39773  hdmapval3N  39779  otiunsndisjX  44658