Theorem s2eqd 12489

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 12292	. . 3 |- ( ph -> <" A "> = <" N "> )
3		s2eqd.2	. . . 4 |- ( ph -> B = O )
4	3	s1eqd 12292	. . 3 |- ( ph -> <" B "> = <" O "> )
5	2, 4	oveq12d 6109	. 2 |- ( ph -> ( <" A "> concat <" B "> ) = ( <" N "> concat <" O "> ) )
6		df-s2 12475	. 2 |- <" A B "> = ( <" A "> concat <" B "> )
7		df-s2 12475	. 2 |- <" N O "> = ( <" N "> concat <" O "> )
8	5, 6, 7	3eqtr4g 2500	1 |- ( ph -> <" A B "> = <" N O "> )

Syntax hints:   -> wi 4   = wceq 1369  (class class class)co 6091   concat cconcat 12223   <"cs1 12224   <"cs2 12468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-iota 5381  df-fv 5426  df-ov 6094  df-s1 12232  df-s2 12475

This theorem is referenced by:  s3eqd  12490  swrd2lsw  12552  efgi  16216  efgi0  16217  efgi1  16218  efgtf  16219  efgtval  16220  efgval2  16221  frgpuplem  16269