Theorem s3eq2 15032

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 2764	. 2 ⊢ (𝐵 = 𝐷 → 𝐴 = 𝐴)
2		id 23	. 2 ⊢ (𝐵 = 𝐷 → 𝐵 = 𝐷)
3		eqidd 2764	. 2 ⊢ (𝐵 = 𝐷 → 𝐶 = 𝐶)
4	1, 2, 3	s3eqd 15026	1 ⊢ (𝐵 = 𝐷 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐴𝐷𝐶”⟩)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  ⟨“cs3 15004

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-iota 6500  df-fv 6554  df-ov 7430  df-s1 14753  df-s2 15010  df-s3 15011

This theorem is used by:  s3rex  15113  tgcgrxfr  28993  isperp2  29202  elcgrabasi  29387  elwwlks2ons3  30556  frgr2wwlk1  30942  frgr2wwlkeqm  30944  fusgr2wsp2nb  30947