Theorem eqvin.l2 931

Description: A variable elimination law for equality. Lemma 15 of [Monk2] p. 109.

Assertion

Ref	Expression
eqvin.l2	|- (E.z(x = z /\ z = y) -> x = y)

Distinct variable group(s):   x,z   y,z

Proof of Theorem eqvin.l2

Step	Hyp	Ref	Expression
1		ax-17 925	. 2 |- (x = y -> A.z x = y)
2		eqt 814	. . 3 |- (x = z -> (z = y -> x = y))
3	2	imp 277	. 2 |- ((x = z /\ z = y) -> x = y)
4	1, 3	19.23ai 746	1 |- (E.z(x = z /\ z = y) -> x = y)

Syntax hints:   -> wi 2   /\ wa 196  E.wex 678   = weq 797

This theorem is referenced by:  eqvin 932

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-8 798  ax-9 799  ax-12 802  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

metamath.org