Theorem s1eqd 14629

Description: Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)

Hypothesis

Ref	Expression
s1eqd.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
s1eqd	⊢ (𝜑 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)

Proof of Theorem s1eqd

Step	Hyp	Ref	Expression
1		s1eqd.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		s1eq 14628	. 2 ⊢ (𝐴 = 𝐵 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)
3	1, 2	syl 18	1 ⊢ (𝜑 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)

Syntax hints:   → wi 4   = wceq 1563  ⟨“cs1 14623

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 4869  df-br 5106  df-iota 6481  df-fv 6533  df-s1 14624

This theorem is referenced by:  s1prc  14632  ccat1st1st  14656  swrds1  14694  swrdlsw  14695  reuccatpfxs1lem  14773  s2eqd  14890  s3eqd  14891  s4eqd  14892  s5eqd  14893  s6eqd  14894  s7eqd  14895  s8eqd  14896  frmdgsum  18911  psgnunilem5  19555  efgredlemc  19806  vrgpval  19828  vrgpinv  19830  frgpup2  19837  frgpup3lem  19838  pfx1s2  33172  pfxlsw2ccat  33183  ccatws1f1olast  33185  wrdpmtrlast  33326  1arithidomlem2  33743  iwrdsplit  34694  sseqval  34695  sseqf  34699  sseqp1  34702  signsvtn0  34874  signstfveq0  34881  mrsubcv  35873