Theorem s3eq2 14564

Description: Equality theorem for a length 3 word for the second symbol. (Contributed by AV, 4-Jan-2022.)

Assertion

Ref	Expression
s3eq2	⊢ (𝐵 = 𝐷 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐴𝐷𝐶”⟩)

Proof of Theorem s3eq2

Step	Hyp	Ref	Expression
1		eqidd 2740	. 2 ⊢ (𝐵 = 𝐷 → 𝐴 = 𝐴)
2		id 22	. 2 ⊢ (𝐵 = 𝐷 → 𝐵 = 𝐷)
3		eqidd 2740	. 2 ⊢ (𝐵 = 𝐷 → 𝐶 = 𝐶)
4	1, 2, 3	s3eqd 14558	1 ⊢ (𝐵 = 𝐷 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐴𝐷𝐶”⟩)

Syntax hints:   → wi 4   = wceq 1541  ⟨“cs3 14536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-iota 6388  df-fv 6438  df-ov 7271  df-s1 14282  df-s2 14542  df-s3 14543

This theorem is referenced by:  tgcgrxfr  26860  isperp2  27057  elwwlks2ons3  28299  frgr2wwlk1  28672  frgr2wwlkeqm  28674  fusgr2wsp2nb  28677