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Theorem oteq2 4635
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
StepHypRef Expression
1 opeq2 4626 . . 3 (𝐴 = 𝐵 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)
21opeq1d 4631 . 2 (𝐴 = 𝐵 → ⟨⟨𝐶, 𝐴⟩, 𝐷⟩ = ⟨⟨𝐶, 𝐵⟩, 𝐷⟩)
3 df-ot 4408 . 2 𝐶, 𝐴, 𝐷⟩ = ⟨⟨𝐶, 𝐴⟩, 𝐷
4 df-ot 4408 . 2 𝐶, 𝐵, 𝐷⟩ = ⟨⟨𝐶, 𝐵⟩, 𝐷
52, 3, 43eqtr4g 2886 1 (𝐴 = 𝐵 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1656  cop 4405  cotp 4407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-ot 4408
This theorem is referenced by:  oteq2d  4638  efgi  18490  efgtf  18493  efgtval  18494
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