Theorem equtr2 1688

Description: A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
equtr2	⊢ ((x = z ∧ y = z) → x = y)

Proof of Theorem equtr2

Step	Hyp	Ref	Expression
1		equtrr 1683	. . 3 ⊢ (z = y → (x = z → x = y))
2	1	equcoms 1681	. 2 ⊢ (y = z → (x = z → x = y))
3	2	impcom 419	1 ⊢ ((x = z ∧ y = z) → x = y)

Syntax hints:   → wi 4   ∧ wa 358

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

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

This theorem is referenced by:  mo  2226  2mo  2282  euequ1  2292