Theorem eqtr3 2372

Description: A transitive law for class equality. (Contributed by NM, 20-May-2005.)

Assertion

Ref	Expression
eqtr3	⊢ ((A = C ∧ B = C) → A = B)

Proof of Theorem eqtr3

Step	Hyp	Ref	Expression
1		eqcom 2355	. 2 ⊢ (B = C ↔ C = B)
2		eqtr 2370	. 2 ⊢ ((A = C ∧ C = B) → A = B)
3	1, 2	sylan2b 461	1 ⊢ ((A = C ∧ B = C) → A = B)

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:  eueq  3009  euind  3024  reuind  3040  phialllem1  4617  xpcan  5058  xpcan2  5059  foco  5280  xpassen  6058  enmap2lem4  6067  enmap1lem4  6073  ncspw1eu  6160