Theorem s5eqd 11528

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 11527	. . 3 |- ( ph -> <" A B C D "> = <" N O P Q "> )
6		s5eqd.5	. . . 4 |- ( ph -> E = R )
7	6	s1eqd 11371	. . 3 |- ( ph -> <" E "> = <" R "> )
8	5, 7	oveq12d 6097	. 2 |- ( ph -> ( <" A B C D "> ++ <" E "> ) = ( <" N O P Q "> ++ <" R "> ) )
9		df-s5 11514	. 2 |- <" A B C D E "> = ( <" A B C D "> ++ <" E "> )
10		df-s5 11514	. 2 |- <" N O P Q R "> = ( <" N O P Q "> ++ <" R "> )
11	8, 9, 10	3eqtr4g 2296	1 |- ( ph -> <" A B C D E "> = <" N O P Q R "> )

Syntax hints:   -> wi 4   = wceq 1402  (class class class)co 6079   ++ cconcat 11341   <"cs1 11366   <"cs4 11506   <"cs5 11507

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  df-s4 11513  df-s5 11514

This theorem is referenced by:  s6eqd  11529