Theorem sneqd 3454

Description: Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)

Hypothesis

Ref	Expression
sneqd.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
sneqd	⊢ (𝜑 → {𝐴} = {𝐵})

Proof of Theorem sneqd

Step	Hyp	Ref	Expression
1		sneqd.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		sneq 3452	. 2 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
3	1, 2	syl 14	1 ⊢ (𝜑 → {𝐴} = {𝐵})

Syntax hints:   → wi 4   = wceq 1289  {csn 3441

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-sn 3447

This theorem is referenced by:  dmsnsnsng  4895  cnvsng  4903  ressn  4958  f1osng  5278  fsng  5454  fnressn  5467  fvsng  5477  2nd1st  5932  dfmpt2  5970  cnvf1olem  5971  tpostpos  6011  tfrlemi1  6079  tfr1onlemaccex  6095  tfrcllemaccex  6108  en1bg  6497  mapsnen  6508  xpassen  6526  fztp  9459  fzsuc2  9460  fseq1p1m1  9475  fseq1m1p1  9476  zfz1isolemsplit  10208  zfz1isolem1  10210  fsumm1  10773  divalgmod  11009