Theorem s2eqd 13018

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 12793	. . 3 |- ( ph -> <" A "> = <" N "> )
3		s2eqd.2	. . . 4 |- ( ph -> B = O )
4	3	s1eqd 12793	. . 3 |- ( ph -> <" B "> = <" O "> )
5	2, 4	oveq12d 6326	. 2 |- ( ph -> ( <" A "> ++ <" B "> ) = ( <" N "> ++ <" O "> ) )
6		df-s2 13003	. 2 |- <" A B "> = ( <" A "> ++ <" B "> )
7		df-s2 13003	. 2 |- <" N O "> = ( <" N "> ++ <" O "> )
8	5, 6, 7	3eqtr4g 2530	1 |- ( ph -> <" A B "> = <" N O "> )

Syntax hints:   -> wi 4   = wceq 1452  (class class class)co 6308   ++ cconcat 12705   <"cs1 12706   <"cs2 12996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-iota 5553  df-fv 5597  df-ov 6311  df-s1 12714  df-s2 13003

This theorem is referenced by:  s3eqd  13019  wrdl2exs2  13097  swrd2lsw  13102  efgi  17447  efgi0  17448  efgi1  17449  efgtf  17450  efgtval  17451  efgval2  17452  frgpuplem  17500