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Theorem equvini 1688
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
equvini  |-  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) )

Proof of Theorem equvini
StepHypRef Expression
1 ax12or 1448 . 2  |-  ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  A. z ( x  =  y  ->  A. z  x  =  y )
) )
2 equcomi 1637 . . . . . . 7  |-  ( z  =  x  ->  x  =  z )
32alimi 1389 . . . . . 6  |-  ( A. z  z  =  x  ->  A. z  x  =  z )
4 a9e 1631 . . . . . 6  |-  E. z 
z  =  y
53, 4jctir 306 . . . . 5  |-  ( A. z  z  =  x  ->  ( A. z  x  =  z  /\  E. z  z  =  y
) )
65a1d 22 . . . 4  |-  ( A. z  z  =  x  ->  ( x  =  y  ->  ( A. z  x  =  z  /\  E. z  z  =  y ) ) )
7 19.29 1556 . . . 4  |-  ( ( A. z  x  =  z  /\  E. z 
z  =  y )  ->  E. z ( x  =  z  /\  z  =  y ) )
86, 7syl6 33 . . 3  |-  ( A. z  z  =  x  ->  ( x  =  y  ->  E. z ( x  =  z  /\  z  =  y ) ) )
9 a9e 1631 . . . . . . . 8  |-  E. z 
z  =  x
102eximi 1536 . . . . . . . 8  |-  ( E. z  z  =  x  ->  E. z  x  =  z )
119, 10ax-mp 7 . . . . . . 7  |-  E. z  x  =  z
12112a1i 27 . . . . . 6  |-  ( A. z  z  =  y  ->  ( x  =  y  ->  E. z  x  =  z ) )
1312anc2ri 323 . . . . 5  |-  ( A. z  z  =  y  ->  ( x  =  y  ->  ( E. z  x  =  z  /\  A. z  z  =  y ) ) )
14 19.29r 1557 . . . . 5  |-  ( ( E. z  x  =  z  /\  A. z 
z  =  y )  ->  E. z ( x  =  z  /\  z  =  y ) )
1513, 14syl6 33 . . . 4  |-  ( A. z  z  =  y  ->  ( x  =  y  ->  E. z ( x  =  z  /\  z  =  y ) ) )
16 ax-8 1440 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
x  =  y  -> 
z  =  y ) )
1716anc2li 322 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
x  =  y  -> 
( x  =  z  /\  z  =  y ) ) )
1817equcoms 1641 . . . . . . . . . 10  |-  ( z  =  x  ->  (
x  =  y  -> 
( x  =  z  /\  z  =  y ) ) )
1918com12 30 . . . . . . . . 9  |-  ( x  =  y  ->  (
z  =  x  -> 
( x  =  z  /\  z  =  y ) ) )
2019alimi 1389 . . . . . . . 8  |-  ( A. z  x  =  y  ->  A. z ( z  =  x  ->  (
x  =  z  /\  z  =  y )
) )
21 exim 1535 . . . . . . . 8  |-  ( A. z ( z  =  x  ->  ( x  =  z  /\  z  =  y ) )  ->  ( E. z 
z  =  x  ->  E. z ( x  =  z  /\  z  =  y ) ) )
2220, 21syl 14 . . . . . . 7  |-  ( A. z  x  =  y  ->  ( E. z  z  =  x  ->  E. z
( x  =  z  /\  z  =  y ) ) )
239, 22mpi 15 . . . . . 6  |-  ( A. z  x  =  y  ->  E. z ( x  =  z  /\  z  =  y ) )
2423imim2i 12 . . . . 5  |-  ( ( x  =  y  ->  A. z  x  =  y )  ->  (
x  =  y  ->  E. z ( x  =  z  /\  z  =  y ) ) )
2524sps 1475 . . . 4  |-  ( A. z ( x  =  y  ->  A. z  x  =  y )  ->  ( x  =  y  ->  E. z ( x  =  z  /\  z  =  y ) ) )
2615, 25jaoi 671 . . 3  |-  ( ( A. z  z  =  y  \/  A. z
( x  =  y  ->  A. z  x  =  y ) )  -> 
( x  =  y  ->  E. z ( x  =  z  /\  z  =  y ) ) )
278, 26jaoi 671 . 2  |-  ( ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  A. z ( x  =  y  ->  A. z  x  =  y )
) )  ->  (
x  =  y  ->  E. z ( x  =  z  /\  z  =  y ) ) )
281, 27ax-mp 7 1  |-  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 664   A.wal 1287    = wceq 1289   E.wex 1426
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 665  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  sbequi  1767  equvin  1791
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