Theorem s1eqd 14551

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 14550	. 2 ⊢ (𝐴 = 𝐵 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)
3	1, 2	syl 17	1 ⊢ (𝜑 → ⟨“𝐴”⟩ = ⟨“𝐵”⟩)

Syntax hints:   → wi 4   = wceq 1542  ⟨“cs1 14545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-s1 14546

This theorem is referenced by:  s1prc  14554  ccat1st1st  14578  swrds1  14616  swrdlsw  14617  reuccatpfxs1lem  14696  s2eqd  14814  s3eqd  14815  s4eqd  14816  s5eqd  14817  s6eqd  14818  s7eqd  14819  s8eqd  14820  frmdgsum  18743  psgnunilem5  19362  efgredlemc  19613  vrgpval  19635  vrgpinv  19637  frgpup2  19644  frgpup3lem  19645  pfx1s2  32105  pfxlsw2ccat  32116  iwrdsplit  33386  sseqval  33387  sseqf  33391  sseqp1  33394  signsvtn0  33581  signstfveq0  33588  mrsubcv  34501