Theorem s1eqd 14526

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 14525	. 2 ⊢ (𝐴 = 𝐵 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)
3	1, 2	syl 17	1 ⊢ (𝜑 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)

Syntax hints:   → wi 4   = wceq 1542  ⟨“cs1 14520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6446  df-fv 6498  df-s1 14521

This theorem is referenced by:  s1prc  14529  ccat1st1st  14553  swrds1  14591  swrdlsw  14592  reuccatpfxs1lem  14670  s2eqd  14787  s3eqd  14788  s4eqd  14789  s5eqd  14790  s6eqd  14791  s7eqd  14792  s8eqd  14793  frmdgsum  18788  psgnunilem5  19427  efgredlemc  19678  vrgpval  19700  vrgpinv  19702  frgpup2  19709  frgpup3lem  19710  pfx1s2  33004  pfxlsw2ccat  33015  ccatws1f1olast  33017  wrdpmtrlast  33159  1arithidomlem2  33601  iwrdsplit  34537  sseqval  34538  sseqf  34542  sseqp1  34545  signsvtn0  34720  signstfveq0  34727  mrsubcv  35698