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Theorem cleljustALT2 2101
Description: Alternate proof of cleljust 1906. Compared with cleljustALT 2100, it uses nfv 1772 followed by equsexv 2077 instead of ax-5 1769 followed by equsexhv 2078, so it uses the idiom  F/ x ph instead of  ph  ->  A. x ph to express non-freeness. This style is generally preferred for later theorems. (Contributed by NM, 28-Jan-2004.) (Revised by Mario Carneiro, 21-Dec-2016.) (Revised by BJ, 29-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
cleljustALT2  |-  ( x  e.  y  <->  E. z
( z  =  x  /\  z  e.  y ) )
Distinct variable groups:    x, z    y, z

Proof of Theorem cleljustALT2
StepHypRef Expression
1 nfv 1772 . . 3  |-  F/ z  x  e.  y
2 elequ1 1905 . . 3  |-  ( z  =  x  ->  (
z  e.  y  <->  x  e.  y ) )
31, 2equsexv 2077 . 2  |-  ( E. z ( z  =  x  /\  z  e.  y )  <->  x  e.  y )
43bicomi 207 1  |-  ( x  e.  y  <->  E. z
( z  =  x  /\  z  e.  y ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 375   E.wex 1674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-10 1926  ax-12 1944
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675  df-nf 1679
This theorem is referenced by: (None)
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