Theorem s1eqd 14537

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 14536	. 2 ⊢ (𝐴 = 𝐵 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)
3	1, 2	syl 17	1 ⊢ (𝜑 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)

Syntax hints:   → wi 4   = wceq 1542  ⟨“cs1 14531

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 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-s1 14532

This theorem is referenced by:  s1prc  14540  ccat1st1st  14564  swrds1  14602  swrdlsw  14603  reuccatpfxs1lem  14681  s2eqd  14798  s3eqd  14799  s4eqd  14800  s5eqd  14801  s6eqd  14802  s7eqd  14803  s8eqd  14804  frmdgsum  18799  psgnunilem5  19435  efgredlemc  19686  vrgpval  19708  vrgpinv  19710  frgpup2  19717  frgpup3lem  19718  pfx1s2  33031  pfxlsw2ccat  33042  ccatws1f1olast  33044  wrdpmtrlast  33186  1arithidomlem2  33628  iwrdsplit  34564  sseqval  34565  sseqf  34569  sseqp1  34572  signsvtn0  34747  signstfveq0  34754  mrsubcv  35723