Theorem bnj1083

Description: Technical lemma for bnj69 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj1083.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
	bnj1083.8	|- K = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
Assertion	bnj1083	|- ( f e. K <-> E. n ch )

Proof

Step	Hyp	Ref	Expression
1		bnj1083.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
2		bnj1083.8	|- K = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
3		df-rex	|- ( E. n e. D ( f Fn n /\ ph /\ ps ) <-> E. n ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) )
4	2	abeq2i	|- ( f e. K <-> E. n e. D ( f Fn n /\ ph /\ ps ) )
5		bnj252	|- ( ( n e. D /\ f Fn n /\ ph /\ ps ) <-> ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) )
6	1 5	bitri	|- ( ch <-> ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) )
7	6	exbii	|- ( E. n ch <-> E. n ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) )
8	3 4 7	3bitr4i	|- ( f e. K <-> E. n ch )