Theorem s3eqd 11526

Description: Equality theorem for a length 3 word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Hypotheses

Ref	Expression
s2eqd.1	|- ( ph -> A = N )
s2eqd.2	|- ( ph -> B = O )
s3eqd.3	|- ( ph -> C = P )

Assertion

Ref	Expression
s3eqd	|- ( ph -> <" A B C "> = <" N O P "> )

Proof of Theorem s3eqd

Step	Hyp	Ref	Expression
1		s2eqd.1	. . . 4 |- ( ph -> A = N )
2		s2eqd.2	. . . 4 |- ( ph -> B = O )
3	1, 2	s2eqd 11525	. . 3 |- ( ph -> <" A B "> = <" N O "> )
4		s3eqd.3	. . . 4 |- ( ph -> C = P )
5	4	s1eqd 11371	. . 3 |- ( ph -> <" C "> = <" P "> )
6	3, 5	oveq12d 6097	. 2 |- ( ph -> ( <" A B "> ++ <" C "> ) = ( <" N O "> ++ <" P "> ) )
7		df-s3 11512	. 2 |- <" A B C "> = ( <" A B "> ++ <" C "> )
8		df-s3 11512	. 2 |- <" N O P "> = ( <" N O "> ++ <" P "> )
9	6, 7, 8	3eqtr4g 2296	1 |- ( ph -> <" A B C "> = <" N O P "> )

Syntax hints:   -> wi 4   = wceq 1402  (class class class)co 6079   ++ cconcat 11341   <"cs1 11366   <"cs2 11504   <"cs3 11505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-iota 5335  df-fv 5383  df-ov 6082  df-s1 11367  df-s2 11511  df-s3 11512

This theorem is referenced by:  s4eqd  11527  s3eq2  11532