Theorem s1eqd 14573

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 14572	. 2 ⊢ (𝐴 = 𝐵 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)
3	1, 2	syl 17	1 ⊢ (𝜑 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)

Syntax hints:   → wi 4   = wceq 1540  ⟨“cs1 14567

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-s1 14568

This theorem is referenced by:  s1prc  14576  ccat1st1st  14600  swrds1  14638  swrdlsw  14639  reuccatpfxs1lem  14718  s2eqd  14836  s3eqd  14837  s4eqd  14838  s5eqd  14839  s6eqd  14840  s7eqd  14841  s8eqd  14842  frmdgsum  18796  psgnunilem5  19431  efgredlemc  19682  vrgpval  19704  vrgpinv  19706  frgpup2  19713  frgpup3lem  19714  pfx1s2  32867  pfxlsw2ccat  32879  ccatws1f1olast  32881  wrdpmtrlast  33057  1arithidomlem2  33514  iwrdsplit  34385  sseqval  34386  sseqf  34390  sseqp1  34393  signsvtn0  34568  signstfveq0  34575  mrsubcv  35504