Theorem eqtr2 2107

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 2091	. 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
2		eqtr 2106	. 2 ⊢ ((𝐵 = 𝐴 ∧ 𝐴 = 𝐶) → 𝐵 = 𝐶)
3	1, 2	sylanb 279	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 = 𝐶) → 𝐵 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1290

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-4 1446  ax-17 1465  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-cleq 2082

This theorem is referenced by:  eqvinc  2743  eqvincg  2744  moop2  4089  reusv3i  4296  relop  4601  f0rn0  5220  fliftfun  5591  th3qlem1  6410  enq0ref  7055  enq0tr  7056  genpdisj  7145  addlsub  7911  fsum2dlemstep  10891  0dvds  11157  cncongr1  11426