Theorem eqtr2 2196

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 2179	. 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
2		eqtr 2195	. 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 1353

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 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170

This theorem is referenced by:  eqvinc  2860  eqvincg  2861  moop2  4249  reusv3i  4457  relop  4774  f0rn0  5407  fliftfun  5792  th3qlem1  6632  enq0ref  7427  enq0tr  7428  genpdisj  7517  addlsub  8321  fsum2dlemstep  11433  0dvds  11809  cncongr1  12093