Theorem sneqd 3747

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 3745	. 2 ⊢ (A = B → {A} = {B})
3	1, 2	syl 15	1 ⊢ (φ → {A} = {B})

Syntax hints:   → wi 4   = wceq 1642  {csn 3738

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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-sn 3742

This theorem is referenced by:  elp6  4264  pw1eqadj  4333  elimapw13  4947  dmsnopg  5067  fnunsn  5191  fsng  5434  fnressn  5439  fvsng  5447  freceq12  6312