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Theorem s1eqd 13943
Description: Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)
Hypothesis
Ref Expression
s1eqd.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
s1eqd (𝜑 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)

Proof of Theorem s1eqd
StepHypRef Expression
1 s1eqd.1 . 2 (𝜑𝐴 = 𝐵)
2 s1eq 13942 . 2 (𝐴 = 𝐵 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)
31, 2syl 17 1 (𝜑 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  ⟨“cs1 13937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-s1 13938
This theorem is referenced by:  s1prc  13946  ccat1st1st  13972  swrds1  14016  swrdlsw  14017  reuccatpfxs1lem  14096  s2eqd  14213  s3eqd  14214  s4eqd  14215  s5eqd  14216  s6eqd  14217  s7eqd  14218  s8eqd  14219  frmdgsum  18015  psgnunilem5  18551  efgredlemc  18800  vrgpval  18822  vrgpinv  18824  frgpup2  18831  frgpup3lem  18832  pfx1s2  30542  pfxlsw2ccat  30553  iwrdsplit  31544  sseqval  31545  sseqf  31549  sseqp1  31552  signsvtn0  31739  signstfveq0  31746  mrsubcv  32654
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