Theorem s3eq2 13615

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

Assertion

Ref	Expression
s3eq2	|- ( B = D -> <" A B C "> = <" A D C "> )

Proof of Theorem s3eq2

Step	Hyp	Ref	Expression
1		eqidd 2623	. 2 |- ( B = D -> A = A )
2		id 22	. 2 |- ( B = D -> B = D )
3		eqidd 2623	. 2 |- ( B = D -> C = C )
4	1, 2, 3	s3eqd 13609	1 |- ( B = D -> <" A B C "> = <" A D C "> )

Syntax hints:   -> wi 4   = wceq 1483   <"cs3 13587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-s1 13302  df-s2 13593  df-s3 13594

This theorem is referenced by:  frgr2wwlk1  27193  frgr2wwlkeqm  27195  fusgr2wsp2nb  27198