MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oteq1 Structured version   Visualization version   GIF version

Theorem oteq1 4343
Description: Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
Assertion
Ref Expression
oteq1 (𝐴 = 𝐵 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)

Proof of Theorem oteq1
StepHypRef Expression
1 opeq1 4334 . . 3 (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
21opeq1d 4340 . 2 (𝐴 = 𝐵 → ⟨⟨𝐴, 𝐶⟩, 𝐷⟩ = ⟨⟨𝐵, 𝐶⟩, 𝐷⟩)
3 df-ot 4133 . 2 𝐴, 𝐶, 𝐷⟩ = ⟨⟨𝐴, 𝐶⟩, 𝐷
4 df-ot 4133 . 2 𝐵, 𝐶, 𝐷⟩ = ⟨⟨𝐵, 𝐶⟩, 𝐷
52, 3, 43eqtr4g 2668 1 (𝐴 = 𝐵 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  cop 4130  cotp 4132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-ot 4133
This theorem is referenced by:  oteq1d  4346  otiunsndisj  4896  efgi  17904  efgtf  17907  efgtval  17908  mapdh9a  35921  mapdh9aOLDN  35922  hdmapval2  35966  otiunsndisjX  40152
  Copyright terms: Public domain W3C validator