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Theorem equvini 1732
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 1491 . 2  |-  ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  A. z ( x  =  y  ->  A. z  x  =  y )
) )
2 equcomi 1681 . . . . . . 7  |-  ( z  =  x  ->  x  =  z )
32alimi 1432 . . . . . 6  |-  ( A. z  z  =  x  ->  A. z  x  =  z )
4 a9e 1675 . . . . . 6  |-  E. z 
z  =  y
53, 4jctir 311 . . . . 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 1600 . . . 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 1675 . . . . . . . 8  |-  E. z 
z  =  x
102eximi 1580 . . . . . . . 8  |-  ( E. z  z  =  x  ->  E. z  x  =  z )
119, 10ax-mp 5 . . . . . . 7  |-  E. z  x  =  z
12112a1i 27 . . . . . 6  |-  ( A. z  z  =  y  ->  ( x  =  y  ->  E. z  x  =  z ) )
1312anc2ri 328 . . . . 5  |-  ( A. z  z  =  y  ->  ( x  =  y  ->  ( E. z  x  =  z  /\  A. z  z  =  y ) ) )
14 19.29r 1601 . . . . 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 1483 . . . . . . . . . . . 12  |-  ( x  =  z  ->  (
x  =  y  -> 
z  =  y ) )
1716anc2li 327 . . . . . . . . . . 11  |-  ( x  =  z  ->  (
x  =  y  -> 
( x  =  z  /\  z  =  y ) ) )
1817equcoms 1685 . . . . . . . . . 10  |-  ( z  =  x  ->  (
x  =  y  -> 
( x  =  z  /\  z  =  y ) ) )
1918com12 30 . . . . . . . . 9  |-  ( x  =  y  ->  (
z  =  x  -> 
( x  =  z  /\  z  =  y ) ) )
2019alimi 1432 . . . . . . . 8  |-  ( A. z  x  =  y  ->  A. z ( z  =  x  ->  (
x  =  z  /\  z  =  y )
) )
21 exim 1579 . . . . . . . 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 1518 . . . 4  |-  ( A. z ( x  =  y  ->  A. z  x  =  y )  ->  ( x  =  y  ->  E. z ( x  =  z  /\  z  =  y ) ) )
2615, 25jaoi 706 . . 3  |-  ( ( A. z  z  =  y  \/  A. z
( x  =  y  ->  A. z  x  =  y ) )  -> 
( x  =  y  ->  E. z ( x  =  z  /\  z  =  y ) ) )
278, 26jaoi 706 . 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 5 1  |-  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 698   A.wal 1330    = wceq 1332   E.wex 1469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-i12 1486  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  sbequi  1812  equvin  1836
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