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Theorem equvini 2140
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. See equvin 1853 for a shorter proof requiring fewer axioms when  z is required to be distinct from  x and  y. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 15-Sep-2018.)
Assertion
Ref Expression
equvini  |-  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) )

Proof of Theorem equvini
StepHypRef Expression
1 equtr 1845 . . . 4  |-  ( z  =  x  ->  (
x  =  y  -> 
z  =  y ) )
2 equequ2 1848 . . . . . 6  |-  ( z  =  y  ->  (
x  =  z  <->  x  =  y ) )
32biimprd 226 . . . . 5  |-  ( z  =  y  ->  (
x  =  y  ->  x  =  z )
)
43anc2ri 560 . . . 4  |-  ( z  =  y  ->  (
x  =  y  -> 
( x  =  z  /\  z  =  y ) ) )
51, 4syli 38 . . 3  |-  ( z  =  x  ->  (
x  =  y  -> 
( x  =  z  /\  z  =  y ) ) )
6 19.8a 1907 . . 3  |-  ( ( x  =  z  /\  z  =  y )  ->  E. z ( x  =  z  /\  z  =  y ) )
75, 6syl6 34 . 2  |-  ( z  =  x  ->  (
x  =  y  ->  E. z ( x  =  z  /\  z  =  y ) ) )
8 ax13 2100 . . 3  |-  ( -.  z  =  x  -> 
( x  =  y  ->  A. z  x  =  y ) )
9 ax6e 2055 . . . . 5  |-  E. z 
z  =  y
109, 4eximii 1704 . . . 4  |-  E. z
( x  =  y  ->  ( x  =  z  /\  z  =  y ) )
111019.35i 1733 . . 3  |-  ( A. z  x  =  y  ->  E. z ( x  =  z  /\  z  =  y ) )
128, 11syl6 34 . 2  |-  ( -.  z  =  x  -> 
( x  =  y  ->  E. z ( x  =  z  /\  z  =  y ) ) )
137, 12pm2.61i 167 1  |-  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370   A.wal 1435   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 1838  ax-10 1886  ax-12 1904  ax-13 2052
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664
This theorem is referenced by:  2ax6elem  2242
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