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Theorem equvin 1854
Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 1887, ax-13 2053. (Revised by Wolf Lammen, 10-Jun-2019.)
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 equviniv 1853 . . 3  |-  ( x  =  y  ->  E. z
( x  =  z  /\  y  =  z ) )
2 equcom 1844 . . . . 5  |-  ( y  =  z  <->  z  =  y )
32anbi2i 698 . . . 4  |-  ( ( x  =  z  /\  y  =  z )  <->  ( x  =  z  /\  z  =  y )
)
43exbii 1712 . . 3  |-  ( E. z ( x  =  z  /\  y  =  z )  <->  E. z
( x  =  z  /\  z  =  y ) )
51, 4sylib 199 . 2  |-  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) )
6 equtr 1846 . . . 4  |-  ( x  =  z  ->  (
z  =  y  ->  x  =  y )
)
76imp 430 . . 3  |-  ( ( x  =  z  /\  z  =  y )  ->  x  =  y )
87exlimiv 1766 . 2  |-  ( E. z ( x  =  z  /\  z  =  y )  ->  x  =  y )
95, 8impbii 190 1  |-  ( x  =  y  <->  E. z
( x  =  z  /\  z  =  y ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370   E.wex 1659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660
This theorem is referenced by: (None)
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