Theorem 19.42vvv 1921

Description: Version of 19.42 2105 with three quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 21-Sep-2011.)

Assertion

Ref	Expression
19.42vvv	|- ( E. x E. y E. z ( ph /\ ps ) <-> ( ph /\ E. x E. y E. z ps ) )

Distinct variable groups:   ph, x   ph, y   ph, z

Allowed substitution hints:   ps( x, y, z)

Proof of Theorem 19.42vvv

Step	Hyp	Ref	Expression
1		19.42vv 1920	. . 3 |- ( E. y E. z ( ph /\ ps ) <-> ( ph /\ E. y E. z ps ) )
2	1	exbii 1774	. 2 |- ( E. x E. y E. z ( ph /\ ps ) <-> E. x ( ph /\ E. y E. z ps ) )
3		19.42v 1918	. 2 |- ( E. x ( ph /\ E. y E. z ps ) <-> ( ph /\ E. x E. y E. z ps ) )
4	2, 3	bitri 264	1 |- ( E. x E. y E. z ( ph /\ ps ) <-> ( ph /\ E. x E. y E. z ps ) )

Syntax hints:   <-> wb 196   /\ wa 384   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  ceqsex6v  3248