MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  equvin Unicode version

Theorem equvin 1954
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
StepHypRef Expression
1 equvini 1940 . 2  |-  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) )
2 equtr 1671 . . . 4  |-  ( x  =  z  ->  (
z  =  y  ->  x  =  y )
)
32imp 418 . . 3  |-  ( ( x  =  z  /\  z  =  y )  ->  x  =  y )
43exlimiv 1624 . 2  |-  ( E. z ( x  =  z  /\  z  =  y )  ->  x  =  y )
51, 4impbii 180 1  |-  ( x  =  y  <->  E. z
( x  =  z  /\  z  =  y ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535
  Copyright terms: Public domain W3C validator