Theorem s3eq2 14073

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 2796	. 2 ⊢ (𝐵 = 𝐷 → 𝐴 = 𝐴)
2		id 22	. 2 ⊢ (𝐵 = 𝐷 → 𝐵 = 𝐷)
3		eqidd 2796	. 2 ⊢ (𝐵 = 𝐷 → 𝐶 = 𝐶)
4	1, 2, 3	s3eqd 14067	1 ⊢ (𝐵 = 𝐷 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐴𝐷𝐶”⟩)

Syntax hints:   → wi 4   = wceq 1522  ⟨“cs3 14045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-sn 4477  df-pr 4479  df-op 4483  df-uni 4750  df-br 4967  df-iota 6194  df-fv 6238  df-ov 7024  df-s1 13799  df-s2 14051  df-s3 14052

This theorem is referenced by:  tgcgrxfr  25991  isperp2  26188  elwwlks2ons3  27426  frgr2wwlk1  27805  frgr2wwlkeqm  27807  fusgr2wsp2nb  27810