Theorem s2eqd 12802

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 12588	. . 3 |- ( ph -> <" A "> = <" N "> )
3		s2eqd.2	. . . 4 |- ( ph -> B = O )
4	3	s1eqd 12588	. . 3 |- ( ph -> <" B "> = <" O "> )
5	2, 4	oveq12d 6312	. 2 |- ( ph -> ( <" A "> concat <" B "> ) = ( <" N "> concat <" O "> ) )
6		df-s2 12788	. 2 |- <" A B "> = ( <" A "> concat <" B "> )
7		df-s2 12788	. 2 |- <" N O "> = ( <" N "> concat <" O "> )
8	5, 6, 7	3eqtr4g 2533	1 |- ( ph -> <" A B "> = <" N O "> )

Syntax hints:   -> wi 4   = wceq 1379  (class class class)co 6294   concat cconcat 12512   <"cs1 12513   <"cs2 12781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-iota 5556  df-fv 5601  df-ov 6297  df-s1 12521  df-s2 12788

This theorem is referenced by:  s3eqd  12803  swrd2lsw  12865  efgi  16587  efgi0  16588  efgi1  16589  efgtf  16590  efgtval  16591  efgval2  16592  frgpuplem  16640