Theorem cleljustALT2

Description: Alternate proof of cleljust . Compared with cleljustALT , it uses nfv followed by equsexv instead of ax-5 followed by equsexhv , 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.)

	Ref	Expression
Assertion	cleljustALT2	|- ( x e. y <-> E. z ( z = x /\ z e. y ) )

Proof

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