Theorem bnj1209 33151

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

Hypotheses

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

Assertion

Ref	Expression
bnj1209	|- ( ch -> E. x th )

Distinct variable group:   ch, x

Allowed substitution hints:   ph( x)   th( x)   B( x)

Proof of Theorem bnj1209

Step	Hyp	Ref	Expression
1		bnj1209.1	. . . . 5 |- ( ch -> E. x e. B ph )
2	1	bnj1196 33149	. . . 4 |- ( ch -> E. x ( x e. B /\ ph ) )
3	2	ancli 551	. . 3 |- ( ch -> ( ch /\ E. x ( x e. B /\ ph ) ) )
4		19.42v 1949	. . 3 |- ( E. x ( ch /\ ( x e. B /\ ph ) ) <-> ( ch /\ E. x ( x e. B /\ ph ) ) )
5	3, 4	sylibr 212	. 2 |- ( ch -> E. x ( ch /\ ( x e. B /\ ph ) ) )
6		bnj1209.2	. . 3 |- ( th <-> ( ch /\ x e. B /\ ph ) )
7		3anass 977	. . 3 |- ( ( ch /\ x e. B /\ ph ) <-> ( ch /\ ( x e. B /\ ph ) ) )
8	6, 7	bitri 249	. 2 |- ( th <-> ( ch /\ ( x e. B /\ ph ) ) )
9	5, 8	bnj1198 33150	1 |- ( ch -> E. x th )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   E.wex 1596   e. wcel 1767   E.wrex 2815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803

This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-ex 1597  df-nf 1600  df-rex 2820

This theorem is referenced by:  bnj1501  33419  bnj1523  33423