Theorem s4eqd 11354

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 11353	. . 3 |- ( ph -> <" A B C "> = <" N O P "> )
5		s4eqd.4	. . . 4 |- ( ph -> D = Q )
6	5	s1eqd 11198	. . 3 |- ( ph -> <" D "> = <" Q "> )
7	4, 6	oveq12d 6036	. 2 |- ( ph -> ( <" A B C "> ++ <" D "> ) = ( <" N O P "> ++ <" Q "> ) )
8		df-s4 11340	. 2 |- <" A B C D "> = ( <" A B C "> ++ <" D "> )
9		df-s4 11340	. 2 |- <" N O P Q "> = ( <" N O P "> ++ <" Q "> )
10	7, 8, 9	3eqtr4g 2289	1 |- ( ph -> <" A B C D "> = <" N O P Q "> )

Syntax hints:   -> wi 4   = wceq 1397  (class class class)co 6018   ++ cconcat 11168   <"cs1 11193   <"cs3 11332   <"cs4 11333

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-ov 6021  df-s1 11194  df-s2 11338  df-s3 11339  df-s4 11340

This theorem is referenced by:  s5eqd  11355