Theorem eqtr2 2253

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 2236	. 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
2		eqtr 2252	. 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 1398

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 1496  ax-gen 1498  ax-4 1559  ax-17 1575  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-cleq 2227

This theorem is referenced by:  eqvinc  2943  eqvincg  2944  moop2  4373  reusv3i  4585  relop  4910  f0rn0  5567  fliftfun  5975  th3qlem1  6884  enq0ref  7764  enq0tr  7765  genpdisj  7854  addlsub  8659  wrd2ind  11440  fsum2dlemstep  12145  0dvds  12522  cncongr1  12825  4sqlem12  13125  uhgr2edg  16327  usgredgreu  16337  uspgredg2vtxeu  16339