Theorem s1eqd 11187

Description: Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.)

Hypothesis

Ref	Expression
s1eqd.1	|- ( ph -> A = B )

Assertion

Ref	Expression
s1eqd	|- ( ph -> <" A "> = <" B "> )

Proof of Theorem s1eqd

Step	Hyp	Ref	Expression
1		s1eqd.1	. 2 |- ( ph -> A = B )
2		s1eq 11186	. 2 |- ( A = B -> <" A "> = <" B "> )
3	1, 2	syl 14	1 |- ( ph -> <" A "> = <" B "> )

Syntax hints:   -> wi 4   = wceq 1395   <"cs1 11182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2802  df-un 3202  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-iota 5284  df-fv 5332  df-s1 11183

This theorem is referenced by:  ccat1st1st  11208  swrds1  11239  swrdlsw  11240  reuccatpfxs1lem  11317  s2eqd  11341  s3eqd  11342  s4eqd  11343  s5eqd  11344  s6eqd  11345  s7eqd  11346  s8eqd  11347