Theorem bnj591

Description: Technical lemma for bnj852 . 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
Hypothesis	bnj591.1	|- ( th <-> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) )
Assertion	bnj591	|- ( [. k / j ]. th <-> ( ( n e. D /\ ch /\ ch' ) -> ( f ` k ) = ( g ` k ) ) )

Proof

Step	Hyp	Ref	Expression
1		bnj591.1	|- ( th <-> ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) )
2	1	sbcbii	|- ( [. k / j ]. th <-> [. k / j ]. ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) )
3		vex	|- k e. _V
4		fveq2	|- ( j = k -> ( f ` j ) = ( f ` k ) )
5		fveq2	|- ( j = k -> ( g ` j ) = ( g ` k ) )
6	4 5	eqeq12d	|- ( j = k -> ( ( f ` j ) = ( g ` j ) <-> ( f ` k ) = ( g ` k ) ) )
7	6	imbi2d	|- ( j = k -> ( ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) <-> ( ( n e. D /\ ch /\ ch' ) -> ( f ` k ) = ( g ` k ) ) ) )
8	3 7	sbcie	|- ( [. k / j ]. ( ( n e. D /\ ch /\ ch' ) -> ( f ` j ) = ( g ` j ) ) <-> ( ( n e. D /\ ch /\ ch' ) -> ( f ` k ) = ( g ` k ) ) )
9	2 8	bitri	|- ( [. k / j ]. th <-> ( ( n e. D /\ ch /\ ch' ) -> ( f ` k ) = ( g ` k ) ) )