Theorem sneqb 3876

Description: Biconditional equality for singletons. (Contributed by SF, 14-Jan-2015.)

Hypothesis

Ref	Expression
sneqb.1	|- A e. _V

Assertion

Ref	Expression
sneqb	|- ({A} = {B} <-> A = B)

Proof of Theorem sneqb

Step	Hyp	Ref	Expression
1		sneqb.1	. 2 |- A e. _V
2		sneqbg 3875	. 2 |- (A e. _V -> ({A} = {B} <-> A = B))
3	1, 2	ax-mp 5	1 |- ({A} = {B} <-> A = B)

Syntax hints:   <-> wb 176   = wceq 1642   e. wcel 1710  _Vcvv 2859  {csn 3737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sn 3741

This theorem is referenced by:  snelpw1  4146  otkelins2kg  4253  otkelins3kg  4254  opksnelsik  4265  nnsucelrlem1  4424  eqtfinrelk  4486  oddfinex  4504  nnadjoinlem1  4519  dfop2lem1  4573  setconslem1  4731  setconslem2  4732  funsi  5520  brsnsi  5773  brsnsi1  5775  brsnsi2  5776  funsex  5828  pw1fnex  5852  pw1fnf1o  5855  antisymex  5912  foundex  5914  extex  5915  enpw1  6062  ce2  6192  scancan  6331