Theorem s2eqd 11542

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 11388	. . 3 |- ( ph -> <" A "> = <" N "> )
3		s2eqd.2	. . . 4 |- ( ph -> B = O )
4	3	s1eqd 11388	. . 3 |- ( ph -> <" B "> = <" O "> )
5	2, 4	oveq12d 6103	. 2 |- ( ph -> ( <" A "> ++ <" B "> ) = ( <" N "> ++ <" O "> ) )
6		df-s2 11528	. 2 |- <" A B "> = ( <" A "> ++ <" B "> )
7		df-s2 11528	. 2 |- <" N O "> = ( <" N "> ++ <" O "> )
8	5, 6, 7	3eqtr4g 2296	1 |- ( ph -> <" A B "> = <" N O "> )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402  (class class class)co 6085   ++ cconcat 11358   <"cs1 11383   <"cs2 11521

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385  df-ov 6088  df-s1 11384  df-s2 11528

This theorem is used by:  s3eqd  11543