Theorem eqtr2 2371

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 2355	. 2 ⊢ (A = B ↔ B = A)
2		eqtr 2370	. 2 ⊢ ((B = A ∧ A = C) → B = C)
3	1, 2	sylanb 458	1 ⊢ ((A = B ∧ A = C) → B = C)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346

This theorem is referenced by:  eqvinc  2967  dfxp2  5114  1stfo  5506  2ndfo  5507  swapf1o  5512  brtxp  5784  enprmaplem3  6079  peano4nc  6151  nchoicelem17  6306  fnfrec  6321