Theorem s2eqd 11781

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 11709	. . 3 |- ( ph -> <" A "> = <" N "> )
3		s2eqd.2	. . . 4 |- ( ph -> B = O )
4	3	s1eqd 11709	. . 3 |- ( ph -> <" B "> = <" O "> )
5	2, 4	oveq12d 6058	. 2 |- ( ph -> ( <" A "> concat <" B "> ) = ( <" N "> concat <" O "> ) )
6		df-s2 11767	. 2 |- <" A B "> = ( <" A "> concat <" B "> )
7		df-s2 11767	. 2 |- <" N O "> = ( <" N "> concat <" O "> )
8	5, 6, 7	3eqtr4g 2461	1 |- ( ph -> <" A B "> = <" N O "> )

Syntax hints:   -> wi 4   = wceq 1649  (class class class)co 6040   concat cconcat 11673   <"cs1 11674   <"cs2 11760

This theorem is referenced by:  s3eqd  11782  efgi  15306  efgi0  15307  efgi1  15308  efgtf  15309  efgtval  15310  efgval2  15311  frgpuplem  15359

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421  df-ov 6043  df-s1 11680  df-s2 11767