Theorem s3eq2 15021

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 2762	. 2 ⊢ (𝐵 = 𝐷 → 𝐴 = 𝐴)
2		id 23	. 2 ⊢ (𝐵 = 𝐷 → 𝐵 = 𝐷)
3		eqidd 2762	. 2 ⊢ (𝐵 = 𝐷 → 𝐶 = 𝐶)
4	1, 2, 3	s3eqd 15015	1 ⊢ (𝐵 = 𝐷 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐴𝐷𝐶”⟩)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ⟨“cs3 14993

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6494  df-fv 6546  df-ov 7423  df-s1 14743  df-s2 14999  df-s3 15000

This theorem is used by:  s3rex  15101  tgcgrxfr  28981  isperp2  29190  elcgrabasi  29375  elwwlks2ons3  30544  frgr2wwlk1  30930  frgr2wwlkeqm  30932  fusgr2wsp2nb  30935