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Theorem s1eqd 14639
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 14638 . 2 (𝐴 = 𝐵 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)
31, 2syl 17 1 (𝜑 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  ⟨“cs1 14633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-s1 14634
This theorem is referenced by:  s1prc  14642  ccat1st1st  14666  swrds1  14704  swrdlsw  14705  reuccatpfxs1lem  14784  s2eqd  14902  s3eqd  14903  s4eqd  14904  s5eqd  14905  s6eqd  14906  s7eqd  14907  s8eqd  14908  frmdgsum  18875  psgnunilem5  19512  efgredlemc  19763  vrgpval  19785  vrgpinv  19787  frgpup2  19794  frgpup3lem  19795  pfx1s2  32923  pfxlsw2ccat  32935  ccatws1f1olast  32937  wrdpmtrlast  33113  1arithidomlem2  33564  iwrdsplit  34389  sseqval  34390  sseqf  34394  sseqp1  34397  signsvtn0  34585  signstfveq0  34592  mrsubcv  35515
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