Theorem s2eqd 12806

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 12592	. . 3 |- ( ph -> <" A "> = <" N "> )
3		s2eqd.2	. . . 4 |- ( ph -> B = O )
4	3	s1eqd 12592	. . 3 |- ( ph -> <" B "> = <" O "> )
5	2, 4	oveq12d 6299	. 2 |- ( ph -> ( <" A "> concat <" B "> ) = ( <" N "> concat <" O "> ) )
6		df-s2 12792	. 2 |- <" A B "> = ( <" A "> concat <" B "> )
7		df-s2 12792	. 2 |- <" N O "> = ( <" N "> concat <" O "> )
8	5, 6, 7	3eqtr4g 2509	1 |- ( ph -> <" A B "> = <" N O "> )

Syntax hints:   -> wi 4   = wceq 1383  (class class class)co 6281   concat cconcat 12515   <"cs1 12516   <"cs2 12785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-iota 5541  df-fv 5586  df-ov 6284  df-s1 12524  df-s2 12792

This theorem is referenced by:  s3eqd  12807  swrd2lsw  12869  efgi  16611  efgi0  16612  efgi1  16613  efgtf  16614  efgtval  16615  efgval2  16616  frgpuplem  16664