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Theorem equviniNEW7 29625
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 1693 . . . . . 6  |-  ( z  =  x  ->  x  =  z )
21alimi 1569 . . . . 5  |-  ( A. z  z  =  x  ->  A. z  x  =  z )
3 a9eNEW7 29568 . . . . 5  |-  E. z 
z  =  y
42, 3jctir 526 . . . 4  |-  ( A. z  z  =  x  ->  ( A. z  x  =  z  /\  E. z  z  =  y
) )
54a1d 24 . . 3  |-  ( A. z  z  =  x  ->  ( x  =  y  ->  ( A. z  x  =  z  /\  E. z  z  =  y ) ) )
6 19.29 1607 . . 3  |-  ( ( A. z  x  =  z  /\  E. z 
z  =  y )  ->  E. z ( x  =  z  /\  z  =  y ) )
75, 6syl6 32 . 2  |-  ( A. z  z  =  x  ->  ( x  =  y  ->  E. z ( x  =  z  /\  z  =  y ) ) )
8 a9eNEW7 29568 . . . . . 6  |-  E. z 
z  =  x
98, 1eximii 1588 . . . . 5  |-  E. z  x  =  z
109a1ii 26 . . . 4  |-  ( A. z  z  =  y  ->  ( x  =  y  ->  E. z  x  =  z ) )
1110anc2ri 543 . . 3  |-  ( A. z  z  =  y  ->  ( x  =  y  ->  ( E. z  x  =  z  /\  A. z  z  =  y ) ) )
12 19.29r 1608 . . 3  |-  ( ( E. z  x  =  z  /\  A. z 
z  =  y )  ->  E. z ( x  =  z  /\  z  =  y ) )
1311, 12syl6 32 . 2  |-  ( A. z  z  =  y  ->  ( x  =  y  ->  E. z ( x  =  z  /\  z  =  y ) ) )
14 ioran 478 . . 3  |-  ( -.  ( A. z  z  =  x  \/  A. z  z  =  y
)  <->  ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y ) )
15 nfeqfNEW7 29584 . . . 4  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z  x  =  y )
16 ax-8 1689 . . . . . 6  |-  ( x  =  z  ->  (
x  =  y  -> 
z  =  y ) )
1716anc2li 542 . . . . 5  |-  ( x  =  z  ->  (
x  =  y  -> 
( x  =  z  /\  z  =  y ) ) )
1817equcoms 1695 . . . 4  |-  ( z  =  x  ->  (
x  =  y  -> 
( x  =  z  /\  z  =  y ) ) )
1915, 18spimedNEW7 29608 . . 3  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) ) )
2014, 19sylbi 189 . 2  |-  ( -.  ( A. z  z  =  x  \/  A. z  z  =  y
)  ->  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) ) )
217, 13, 20ecase3 909 1  |-  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 359    /\ wa 360   A.wal 1550   E.wex 1551
This theorem is referenced by:  equvinNEW7  29627  sbequiNEW7  29677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-11 1763  ax-12 1953  ax-7v 29540
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555
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