Theorem equvin 1784

Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
equvin	|- ( x = y <-> E. z ( x = z /\ z = y ) )

Distinct variable groups:   x, z   y, z

Proof of Theorem equvin

Step	Hyp	Ref	Expression
1		equvini 1681	. 2 |- ( x = y -> E. z ( x = z /\ z = y ) )
2		ax-17 1459	. . 3 |- ( x = y -> A. z x = y )
3		equtr 1635	. . . 4 |- ( x = z -> ( z = y -> x = y ) )
4	3	imp 122	. . 3 |- ( ( x = z /\ z = y ) -> x = y )
5	2, 4	exlimih 1524	. 2 |- ( E. z ( x = z /\ z = y ) -> x = y )
6	1, 5	impbii 124	1 |- ( x = y <-> E. z ( x = z /\ z = y ) )

Syntax hints:   /\ wa 102   <-> wb 103   E.wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-i12 1438  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115

This theorem is referenced by: (None)