MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  s3eq2 Structured version   Visualization version   GIF version

Theorem s3eq2 14227
Description: Equality theorem for a length 3 word for the second symbol. (Contributed by AV, 4-Jan-2022.)
Assertion
Ref Expression
s3eq2 (𝐵 = 𝐷 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐴𝐷𝐶”⟩)

Proof of Theorem s3eq2
StepHypRef Expression
1 eqidd 2802 . 2 (𝐵 = 𝐷𝐴 = 𝐴)
2 id 22 . 2 (𝐵 = 𝐷𝐵 = 𝐷)
3 eqidd 2802 . 2 (𝐵 = 𝐷𝐶 = 𝐶)
41, 2, 3s3eqd 14221 1 (𝐵 = 𝐷 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐴𝐷𝐶”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  ⟨“cs3 14199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336  df-ov 7142  df-s1 13945  df-s2 14205  df-s3 14206
This theorem is referenced by:  tgcgrxfr  26316  isperp2  26513  elwwlks2ons3  27745  frgr2wwlk1  28118  frgr2wwlkeqm  28120  fusgr2wsp2nb  28123
  Copyright terms: Public domain W3C validator