Theorem equvin 1911

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 1806	. 2 |- ( x = y -> E. z ( x = z /\ z = y ) )
2		ax-17 1574	. . 3 |- ( x = y -> A. z x = y )
3		equtr 1757	. . . 4 |- ( x = z -> ( z = y -> x = y ) )
4	3	imp 124	. . 3 |- ( ( x = z /\ z = y ) -> x = y )
5	2, 4	exlimih 1641	. 2 |- ( E. z ( x = z /\ z = y ) -> x = y )
6	1, 5	impbii 126	1 |- ( x = y <-> E. z ( x = z /\ z = y ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-i12 1555  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)