Theorem s4eqd 11544

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 11543	. . 3 |- ( ph -> <" A B C "> = <" N O P "> )
5		s4eqd.4	. . . 4 |- ( ph -> D = Q )
6	5	s1eqd 11388	. . 3 |- ( ph -> <" D "> = <" Q "> )
7	4, 6	oveq12d 6103	. 2 |- ( ph -> ( <" A B C "> ++ <" D "> ) = ( <" N O P "> ++ <" Q "> ) )
8		df-s4 11530	. 2 |- <" A B C D "> = ( <" A B C "> ++ <" D "> )
9		df-s4 11530	. 2 |- <" N O P Q "> = ( <" N O P "> ++ <" Q "> )
10	7, 8, 9	3eqtr4g 2296	1 |- ( ph -> <" A B C D "> = <" N O P Q "> )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402  (class class class)co 6085   ++ cconcat 11358   <"cs1 11383   <"cs3 11522   <"cs4 11523

This proof depends on 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 proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385  df-ov 6088  df-s1 11384  df-s2 11528  df-s3 11529  df-s4 11530

This theorem is used by:  s5eqd  11545