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Theorem s1eq 14626
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 6871 . . . 4 (𝐴 = 𝐵 → ( I ‘𝐴) = ( I ‘𝐵))
21opeq2d 4840 . . 3 (𝐴 = 𝐵 → ⟨0, ( I ‘𝐴)⟩ = ⟨0, ( I ‘𝐵)⟩)
32sneqd 4597 . 2 (𝐴 = 𝐵 → {⟨0, ( I ‘𝐴)⟩} = {⟨0, ( I ‘𝐵)⟩})
4 df-s1 14622 . 2 ⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}
5 df-s1 14622 . 2 ⟨“𝐵”⟩ = {⟨0, ( I ‘𝐵)⟩}
63, 4, 53eqtr4g 2825 1 (𝐴 = 𝐵 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1563  {csn 4585  cop 4591   I cid 5545  cfv 6525  0cc0 11088  ⟨“cs1 14621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737
This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-iota 6481  df-fv 6533  df-s1 14622
This theorem is referenced by:  s1eqd  14627  wrdl1exs1  14639  wrdl1s1  14640  ccats1pfxeqrex  14740  wrdind  14747  wrd2ind  14748  reuccatpfxs1lem  14771  reuccatpfxs1  14772  revs1  14790  chninf  18679  vrmdval  18904  frgpup3lem  19835  vdegp1ci  29793  clwwlknonwwlknonb  30362  mrsubcv  35868  mrsubrn  35871  elmrsubrn  35878  mrsubvrs  35880  mvhval  35892
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