Metamath Proof Explorer


Theorem bnj1521

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

Ref Expression
Hypotheses bnj1521.1
|- ( ch -> E. x e. B ph )
bnj1521.2
|- ( th <-> ( ch /\ x e. B /\ ph ) )
bnj1521.3
|- ( ch -> A. x ch )
Assertion bnj1521
|- ( ch -> E. x th )

Proof

Step Hyp Ref Expression
1 bnj1521.1
 |-  ( ch -> E. x e. B ph )
2 bnj1521.2
 |-  ( th <-> ( ch /\ x e. B /\ ph ) )
3 bnj1521.3
 |-  ( ch -> A. x ch )
4 1 bnj1196
 |-  ( ch -> E. x ( x e. B /\ ph ) )
5 4 2 3 bnj1345
 |-  ( ch -> E. x th )