Theorem eqtr2 2250

Description: A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
eqtr2	⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐶) → 𝐵 = 𝐶)

Proof of Theorem eqtr2

Step	Hyp	Ref	Expression
1		eqcom 2233	. 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
2		eqtr 2249	. 2 ⊢ ((𝐵 = 𝐴 ∧ 𝐴 = 𝐶) → 𝐵 = 𝐶)
3	1, 2	sylanb 284	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐶) → 𝐵 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397

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-5 1495  ax-gen 1497  ax-4 1558  ax-17 1574  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224

This theorem is referenced by:  eqvinc  2929  eqvincg  2930  moop2  4344  reusv3i  4556  relop  4880  f0rn0  5531  fliftfun  5936  th3qlem1  6805  enq0ref  7652  enq0tr  7653  genpdisj  7742  addlsub  8548  wrd2ind  11303  fsum2dlemstep  11994  0dvds  12371  cncongr1  12674  4sqlem12  12974  uhgr2edg  16056  usgredgreu  16066  uspgredg2vtxeu  16068