Theorem oteq2 4883

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

Assertion

Ref	Expression
oteq2	⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)

Proof of Theorem oteq2

Step	Hyp	Ref	Expression
1		opeq2 4874	. . 3 ⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)
2	1	opeq1d 4879	. 2 ⊢ (𝐴 = 𝐵 → ⟨⟨𝐶, 𝐴⟩, 𝐷⟩ = ⟨⟨𝐶, 𝐵⟩, 𝐷⟩)
3		df-ot 4635	. 2 ⊢ ⟨𝐶, 𝐴, 𝐷⟩ = ⟨⟨𝐶, 𝐴⟩, 𝐷⟩
4		df-ot 4635	. 2 ⊢ ⟨𝐶, 𝐵, 𝐷⟩ = ⟨⟨𝐶, 𝐵⟩, 𝐷⟩
5	2, 3, 4	3eqtr4g 2802	1 ⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)

Syntax hints:   → wi 4   = wceq 1540  ⟨cop 4632  ⟨cotp 4634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-ot 4635

This theorem is referenced by:  oteq2d  4886  frxp3  8176  xpord3pred  8177  efgi  19737  efgtf  19740  efgtval  19741