Theorem eqtr 2370

Description: Transitive law for class equality. Proposition 4.7(3) of [TakeutiZaring] p. 13. (Contributed by NM, 25-Jan-2004.)

Assertion

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

Proof of Theorem eqtr

Step	Hyp	Ref	Expression
1		eqeq1 2359	. 2 ⊢ (A = B → (A = C ↔ B = C))
2	1	biimpar 471	1 ⊢ ((A = B ∧ B = C) → A = 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:  eqtr2  2371  eqtr3  2372  sylan9eq  2405  eqvinc  2967  uneqdifeq  3639  ider  5944  eqer  5962  ncaddccl  6145  ncdisjeq  6149  nchoicelem17  6306