Theorem bnj1209

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj1209.1	|- ( ch -> E. x e. B ph )
	bnj1209.2	|- ( th <-> ( ch /\ x e. B /\ ph ) )
Assertion	bnj1209	|- ( ch -> E. x th )

Proof

Step	Hyp	Ref	Expression
1		bnj1209.1	|- ( ch -> E. x e. B ph )
2		bnj1209.2	|- ( th <-> ( ch /\ x e. B /\ ph ) )
3	1	bnj1196	|- ( ch -> E. x ( x e. B /\ ph ) )
4	3	ancli	|- ( ch -> ( ch /\ E. x ( x e. B /\ ph ) ) )
5		19.42v	|- ( E. x ( ch /\ ( x e. B /\ ph ) ) <-> ( ch /\ E. x ( x e. B /\ ph ) ) )
6	4 5	sylibr	|- ( ch -> E. x ( ch /\ ( x e. B /\ ph ) ) )
7		3anass	|- ( ( ch /\ x e. B /\ ph ) <-> ( ch /\ ( x e. B /\ ph ) ) )
8	2 7	bitri	|- ( th <-> ( ch /\ ( x e. B /\ ph ) ) )
9	6 8	bnj1198	|- ( ch -> E. x th )