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Theorem equvin 1786
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 1683 . 2  |-  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) )
2 ax-17 1460 . . 3  |-  ( x  =  y  ->  A. z  x  =  y )
3 equtr 1637 . . . 4  |-  ( x  =  z  ->  (
z  =  y  ->  x  =  y )
)
43imp 122 . . 3  |-  ( ( x  =  z  /\  z  =  y )  ->  x  =  y )
52, 4exlimih 1525 . 2  |-  ( E. z ( x  =  z  /\  z  =  y )  ->  x  =  y )
61, 5impbii 124 1  |-  ( x  =  y  <->  E. z
( x  =  z  /\  z  =  y ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103   E.wex 1422
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 663  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
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