Theorem s3eq2 14805

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 2738	. 2 ⊢ (𝐵 = 𝐷 → 𝐴 = 𝐴)
2		id 22	. 2 ⊢ (𝐵 = 𝐷 → 𝐵 = 𝐷)
3		eqidd 2738	. 2 ⊢ (𝐵 = 𝐷 → 𝐶 = 𝐶)
4	1, 2, 3	s3eqd 14799	1 ⊢ (𝐵 = 𝐷 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐴𝐷𝐶”⟩)

Syntax hints:   → wi 4   = wceq 1542  ⟨“cs3 14777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-s1 14532  df-s2 14783  df-s3 14784

This theorem is referenced by:  tgcgrxfr  28602  isperp2  28799  elwwlks2ons3  30040  frgr2wwlk1  30416  frgr2wwlkeqm  30418  fusgr2wsp2nb  30421