Theorem bnj984

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	bnj984.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
	bnj984.11	|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
Assertion	bnj984	|- ( G e. A -> ( G e. B <-> [. G / f ]. E. n ch ) )

Proof

Step	Hyp	Ref	Expression
1		bnj984.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
2		bnj984.11	|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
3	2	eleq2i	|- ( G e. B <-> G e. { f | E. n e. D ( f Fn n /\ ph /\ ps ) } )
4		sbc8g	|- ( G e. A -> ( [. G / f ]. E. n e. D ( f Fn n /\ ph /\ ps ) <-> G e. { f | E. n e. D ( f Fn n /\ ph /\ ps ) } ) )
5	3 4	bitr4id	|- ( G e. A -> ( G e. B <-> [. G / f ]. E. n e. D ( f Fn n /\ ph /\ ps ) ) )
6		df-rex	|- ( E. n e. D ( f Fn n /\ ph /\ ps ) <-> E. n ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) )
7		bnj252	|- ( ( n e. D /\ f Fn n /\ ph /\ ps ) <-> ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) )
8	1 7	bitri	|- ( ch <-> ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) )
9	6 8	bnj133	|- ( E. n e. D ( f Fn n /\ ph /\ ps ) <-> E. n ch )
10	9	sbcbii	|- ( [. G / f ]. E. n e. D ( f Fn n /\ ph /\ ps ) <-> [. G / f ]. E. n ch )
11	5 10	bitrdi	|- ( G e. A -> ( G e. B <-> [. G / f ]. E. n ch ) )