Theorem s2eqd 12485

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 12288	. . 3 |- ( ph -> <" A "> = <" N "> )
3		s2eqd.2	. . . 4 |- ( ph -> B = O )
4	3	s1eqd 12288	. . 3 |- ( ph -> <" B "> = <" O "> )
5	2, 4	oveq12d 6108	. 2 |- ( ph -> ( <" A "> concat <" B "> ) = ( <" N "> concat <" O "> ) )
6		df-s2 12471	. 2 |- <" A B "> = ( <" A "> concat <" B "> )
7		df-s2 12471	. 2 |- <" N O "> = ( <" N "> concat <" O "> )
8	5, 6, 7	3eqtr4g 2498	1 |- ( ph -> <" A B "> = <" N O "> )

Syntax hints:   -> wi 4   = wceq 1364  (class class class)co 6090   concat cconcat 12219   <"cs1 12220   <"cs2 12464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-iota 5378  df-fv 5423  df-ov 6093  df-s1 12228  df-s2 12471

This theorem is referenced by:  s3eqd  12486  swrd2lsw  12548  efgi  16209  efgi0  16210  efgi1  16211  efgtf  16212  efgtval  16213  efgval2  16214  frgpuplem  16262