Theorem oteq3 4885

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 4875	. 2 ⊢ (𝐴 = 𝐵 → ⟨⟨𝐶, 𝐷⟩, 𝐴⟩ = ⟨⟨𝐶, 𝐷⟩, 𝐵⟩)
2		df-ot 4638	. 2 ⊢ ⟨𝐶, 𝐷, 𝐴⟩ = ⟨⟨𝐶, 𝐷⟩, 𝐴⟩
3		df-ot 4638	. 2 ⊢ ⟨𝐶, 𝐷, 𝐵⟩ = ⟨⟨𝐶, 𝐷⟩, 𝐵⟩
4	1, 2, 3	3eqtr4g 2798	1 ⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)

Syntax hints:   → wi 4   = wceq 1542  ⟨cop 4635  ⟨cotp 4637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-ot 4638

This theorem is referenced by:  oteq3d  4888  otsndisj  5520  otiunsndisj  5521  xpord3pred  8138  efgi0  19588  efgi1  19589  mapdhcl  40598  mapdh6dN  40610  mapdh8  40659  mapdh9a  40660  mapdh9aOLDN  40661  hdmap1l6d  40684  hdmapval  40699  hdmapval2  40703  hdmapval3N  40709  otiunsndisjX  45987