Theorem s2eqd 11352

Description: Equality theorem for a doubleton word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Hypotheses

Ref	Expression
s2eqd.1	|- ( ph -> A = N )
s2eqd.2	|- ( ph -> B = O )

Assertion

Ref	Expression
s2eqd	|- ( ph -> <" A B "> = <" N O "> )

Proof of Theorem s2eqd

Step	Hyp	Ref	Expression
1		s2eqd.1	. . . 4 |- ( ph -> A = N )
2	1	s1eqd 11198	. . 3 |- ( ph -> <" A "> = <" N "> )
3		s2eqd.2	. . . 4 |- ( ph -> B = O )
4	3	s1eqd 11198	. . 3 |- ( ph -> <" B "> = <" O "> )
5	2, 4	oveq12d 6036	. 2 |- ( ph -> ( <" A "> ++ <" B "> ) = ( <" N "> ++ <" O "> ) )
6		df-s2 11338	. 2 |- <" A B "> = ( <" A "> ++ <" B "> )
7		df-s2 11338	. 2 |- <" N O "> = ( <" N "> ++ <" O "> )
8	5, 6, 7	3eqtr4g 2289	1 |- ( ph -> <" A B "> = <" N O "> )

Syntax hints:   -> wi 4   = wceq 1397  (class class class)co 6018   ++ cconcat 11168   <"cs1 11193   <"cs2 11331

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

This theorem is referenced by:  s3eqd  11353