Theorem s1eqd 11388

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 11387	. 2 |- ( A = B -> <" A "> = <" B "> )
3	1, 2	syl 14	1 |- ( ph -> <" A "> = <" B "> )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   <"cs1 11383

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-s1 11384

This theorem is used by:  ccat1st1st  11409  swrds1  11440  swrdlsw  11441  reuccatpfxs1lem  11518  s2eqd  11542  s3eqd  11543  s4eqd  11544  s5eqd  11545  s6eqd  11546  s7eqd  11547  s8eqd  11548