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Theorem s1eq 14495
Description: Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Assertion
Ref Expression
s1eq (𝐴 = 𝐵 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)

Proof of Theorem s1eq
StepHypRef Expression
1 fveq2 6847 . . . 4 (𝐴 = 𝐵 → ( I ‘𝐴) = ( I ‘𝐵))
21opeq2d 4842 . . 3 (𝐴 = 𝐵 → ⟨0, ( I ‘𝐴)⟩ = ⟨0, ( I ‘𝐵)⟩)
32sneqd 4603 . 2 (𝐴 = 𝐵 → {⟨0, ( I ‘𝐴)⟩} = {⟨0, ( I ‘𝐵)⟩})
4 df-s1 14491 . 2 ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
5 df-s1 14491 . 2 ⟨“𝐵”⟩ = {⟨0, ( I ‘𝐵)⟩}
63, 4, 53eqtr4g 2802 1 (𝐴 = 𝐵 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  {csn 4591  cop 4597   I cid 5535  cfv 6501  0cc0 11058  ⟨“cs1 14490
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 2708
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 2715  df-cleq 2729  df-clel 2815  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-iota 6453  df-fv 6509  df-s1 14491
This theorem is referenced by:  s1eqd  14496  wrdl1exs1  14508  wrdl1s1  14509  ccats1pfxeqrex  14610  wrdind  14617  wrd2ind  14618  reuccatpfxs1lem  14641  reuccatpfxs1  14642  revs1  14660  vrmdval  18674  frgpup3lem  19566  vdegp1ci  28528  clwwlknonwwlknonb  29092  mrsubcv  34144  mrsubrn  34147  elmrsubrn  34154  mrsubvrs  34156  mvhval  34168
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