Theorem s5eqd 11347

Description: Equality theorem for a length 5 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 )
s4eqd.4	|- ( ph -> D = Q )
s5eqd.5	|- ( ph -> E = R )

Assertion

Ref	Expression
s5eqd	|- ( ph -> <" A B C D E "> = <" N O P Q R "> )

Proof of Theorem s5eqd

Step	Hyp	Ref	Expression
1		s2eqd.1	. . . 4 |- ( ph -> A = N )
2		s2eqd.2	. . . 4 |- ( ph -> B = O )
3		s3eqd.3	. . . 4 |- ( ph -> C = P )
4		s4eqd.4	. . . 4 |- ( ph -> D = Q )
5	1, 2, 3, 4	s4eqd 11346	. . 3 |- ( ph -> <" A B C D "> = <" N O P Q "> )
6		s5eqd.5	. . . 4 |- ( ph -> E = R )
7	6	s1eqd 11190	. . 3 |- ( ph -> <" E "> = <" R "> )
8	5, 7	oveq12d 6031	. 2 |- ( ph -> ( <" A B C D "> ++ <" E "> ) = ( <" N O P Q "> ++ <" R "> ) )
9		df-s5 11333	. 2 |- <" A B C D E "> = ( <" A B C D "> ++ <" E "> )
10		df-s5 11333	. 2 |- <" N O P Q R "> = ( <" N O P Q "> ++ <" R "> )
11	8, 9, 10	3eqtr4g 2287	1 |- ( ph -> <" A B C D E "> = <" N O P Q R "> )

Syntax hints:   -> wi 4   = wceq 1395  (class class class)co 6013   ++ cconcat 11160   <"cs1 11185   <"cs4 11325   <"cs5 11326

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2802  df-un 3202  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-iota 5284  df-fv 5332  df-ov 6016  df-s1 11186  df-s2 11330  df-s3 11331  df-s4 11332  df-s5 11333

This theorem is referenced by:  s6eqd  11348