Theorem equvin 1743

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 1641	. 2 |- ( x = y -> E. z ( x = z /\ z = y ) )
2		ax-17 1419	. . 3 |- ( x = y -> A. z x = y )
3		equtr 1595	. . . 4 |- ( x = z -> ( z = y -> x = y ) )
4	3	imp 115	. . 3 |- ( ( x = z /\ z = y ) -> x = y )
5	2, 4	exlimih 1484	. 2 |- ( E. z ( x = z /\ z = y ) -> x = y )
6	1, 5	impbii 117	1 |- ( x = y <-> E. z ( x = z /\ z = y ) )

Syntax hints:   /\ wa 97   <-> wb 98   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)