Theorem s4eqd 11457

Description: Equality theorem for a length 4 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 )

Assertion

Ref	Expression
s4eqd	|- ( ph -> <" A B C D "> = <" N O P Q "> )

Proof of Theorem s4eqd

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	1, 2, 3	s3eqd 11456	. . 3 |- ( ph -> <" A B C "> = <" N O P "> )
5		s4eqd.4	. . . 4 |- ( ph -> D = Q )
6	5	s1eqd 11301	. . 3 |- ( ph -> <" D "> = <" Q "> )
7	4, 6	oveq12d 6067	. 2 |- ( ph -> ( <" A B C "> ++ <" D "> ) = ( <" N O P "> ++ <" Q "> ) )
8		df-s4 11443	. 2 |- <" A B C D "> = ( <" A B C "> ++ <" D "> )
9		df-s4 11443	. 2 |- <" N O P Q "> = ( <" N O P "> ++ <" Q "> )
10	7, 8, 9	3eqtr4g 2290	1 |- ( ph -> <" A B C D "> = <" N O P Q "> )

Syntax hints:   -> wi 4   = wceq 1398  (class class class)co 6049   ++ cconcat 11271   <"cs1 11296   <"cs3 11435   <"cs4 11436

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2814  df-un 3214  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-iota 5311  df-fv 5359  df-ov 6052  df-s1 11297  df-s2 11441  df-s3 11442  df-s4 11443

This theorem is referenced by:  s5eqd  11458