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Metamath Proof Explorer

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 )