Theorem s1eqd 14529

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 14528	. 2 ⊢ (𝐴 = 𝐵 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)
3	1, 2	syl 17	1 ⊢ (𝜑 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)

Syntax hints:   → wi 4   = wceq 1542  ⟨“cs1 14523

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 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-iota 6449  df-fv 6501  df-s1 14524

This theorem is referenced by:  s1prc  14532  ccat1st1st  14556  swrds1  14594  swrdlsw  14595  reuccatpfxs1lem  14673  s2eqd  14790  s3eqd  14791  s4eqd  14792  s5eqd  14793  s6eqd  14794  s7eqd  14795  s8eqd  14796  frmdgsum  18791  psgnunilem5  19427  efgredlemc  19678  vrgpval  19700  vrgpinv  19702  frgpup2  19709  frgpup3lem  19710  pfx1s2  33002  pfxlsw2ccat  33013  ccatws1f1olast  33015  wrdpmtrlast  33156  1arithidomlem2  33598  iwrdsplit  34525  sseqval  34526  sseqf  34530  sseqp1  34533  signsvtn0  34708  signstfveq0  34715  mrsubcv  35685