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Theorem equviniNEW7 29379
Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require  z to be distinct from  x and  y (making the proof longer). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
equviniNEW7  |-  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) )

Proof of Theorem equviniNEW7
StepHypRef Expression
1 equcomi 1691 . . . . . 6  |-  ( z  =  x  ->  x  =  z )
21alimi 1568 . . . . 5  |-  ( A. z  z  =  x  ->  A. z  x  =  z )
3 a9eNEW7 29324 . . . . 5  |-  E. z 
z  =  y
42, 3jctir 525 . . . 4  |-  ( A. z  z  =  x  ->  ( A. z  x  =  z  /\  E. z  z  =  y
) )
54a1d 23 . . 3  |-  ( A. z  z  =  x  ->  ( x  =  y  ->  ( A. z  x  =  z  /\  E. z  z  =  y ) ) )
6 19.29 1606 . . 3  |-  ( ( A. z  x  =  z  /\  E. z 
z  =  y )  ->  E. z ( x  =  z  /\  z  =  y ) )
75, 6syl6 31 . 2  |-  ( A. z  z  =  x  ->  ( x  =  y  ->  E. z ( x  =  z  /\  z  =  y ) ) )
8 a9eNEW7 29324 . . . . . 6  |-  E. z 
z  =  x
98, 1eximii 1587 . . . . 5  |-  E. z  x  =  z
109a1ii 25 . . . 4  |-  ( A. z  z  =  y  ->  ( x  =  y  ->  E. z  x  =  z ) )
1110anc2ri 542 . . 3  |-  ( A. z  z  =  y  ->  ( x  =  y  ->  ( E. z  x  =  z  /\  A. z  z  =  y ) ) )
12 19.29r 1607 . . 3  |-  ( ( E. z  x  =  z  /\  A. z 
z  =  y )  ->  E. z ( x  =  z  /\  z  =  y ) )
1311, 12syl6 31 . 2  |-  ( A. z  z  =  y  ->  ( x  =  y  ->  E. z ( x  =  z  /\  z  =  y ) ) )
14 ioran 477 . . 3  |-  ( -.  ( A. z  z  =  x  \/  A. z  z  =  y
)  <->  ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y ) )
15 nfeqfNEW7 29340 . . . 4  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z  x  =  y )
16 ax-8 1687 . . . . . 6  |-  ( x  =  z  ->  (
x  =  y  -> 
z  =  y ) )
1716anc2li 541 . . . . 5  |-  ( x  =  z  ->  (
x  =  y  -> 
( x  =  z  /\  z  =  y ) ) )
1817equcoms 1693 . . . 4  |-  ( z  =  x  ->  (
x  =  y  -> 
( x  =  z  /\  z  =  y ) ) )
1915, 18spimedNEW7 29364 . . 3  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) ) )
2014, 19sylbi 188 . 2  |-  ( -.  ( A. z  z  =  x  \/  A. z  z  =  y
)  ->  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) ) )
217, 13, 20ecase3 908 1  |-  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359   A.wal 1549   E.wex 1550
This theorem is referenced by:  equvinNEW7  29381  sbequiNEW7  29430
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761  ax-12 1950  ax-7v 29296
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554
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