Theorem equtr2 1637

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 1636	. . 3 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 → 𝑥 = 𝑦))
2	1	equcoms 1634	. 2 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑧 → 𝑥 = 𝑦))
3	2	impcom 123	1 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦)

Syntax hints:   → wi 4   ∧ wa 102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-gen 1378  ax-ie2 1423  ax-8 1435  ax-17 1459  ax-i9 1463

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  mo23  1982  euequ1  2036