Theorem eqtr2 2055

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 2039	. 2 ⊢ (A = B ↔ B = A)
2		eqtr 2054	. 2 ⊢ ((B = A ∧ A = 𝐶) → B = 𝐶)
3	1, 2	sylanb 268	1 ⊢ ((A = B ∧ A = 𝐶) → B = 𝐶)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-4 1397  ax-17 1416  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-cleq 2030

This theorem is referenced by:  eqvinc  2661  eqvincg  2662  moop2  3979  reusv3i  4157  relop  4429  fliftfun  5379  th3qlem1  6144  enq0ref  6415  enq0tr  6416  genpdisj  6506