Theorem s4eqd 11265

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 11264	. . 3 |- ( ph -> <" A B C "> = <" N O P "> )
5		s4eqd.4	. . . 4 |- ( ph -> D = Q )
6	5	s1eqd 11114	. . 3 |- ( ph -> <" D "> = <" Q "> )
7	4, 6	oveq12d 5987	. 2 |- ( ph -> ( <" A B C "> ++ <" D "> ) = ( <" N O P "> ++ <" Q "> ) )
8		df-s4 11251	. 2 |- <" A B C D "> = ( <" A B C "> ++ <" D "> )
9		df-s4 11251	. 2 |- <" N O P Q "> = ( <" N O P "> ++ <" Q "> )
10	7, 8, 9	3eqtr4g 2265	1 |- ( ph -> <" A B C D "> = <" N O P Q "> )

Syntax hints:   -> wi 4   = wceq 1373  (class class class)co 5969   ++ cconcat 11086   <"cs1 11109   <"cs3 11243   <"cs4 11244

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2779  df-un 3179  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-br 4061  df-iota 5252  df-fv 5299  df-ov 5972  df-s1 11110  df-s2 11249  df-s3 11250  df-s4 11251

This theorem is referenced by:  s5eqd  11266