Theorem cleljustALT2 2186

Description: Alternate proof of cleljust 1998. Compared with cleljustALT 2185, it uses nfv 1843 followed by equsexv 2109 instead of ax-5 1839 followed by equsexhv 2124, 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

Step	Hyp	Ref	Expression
1		nfv 1843	. . 3 |- F/ z x e. y
2		elequ1 1997	. . 3 |- ( z = x -> ( z e. y <-> x e. y ) )
3	1, 2	equsexv 2109	. 2 |- ( E. z ( z = x /\ z e. y ) <-> x e. y )
4	3	bicomi 214	1 |- ( x e. y <-> E. z ( z = x /\ z e. y ) )

Syntax hints:   <-> wb 196   /\ wa 384   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)