Theorem 19.41vvv 1916

Description: Version of 19.41 2103 with three quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995.)

Assertion

Ref	Expression
19.41vvv	|- ( E. x E. y E. z ( ph /\ ps ) <-> ( E. x E. y E. z ph /\ ps ) )

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

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

Proof of Theorem 19.41vvv

Step	Hyp	Ref	Expression
1		19.41vv 1915	. . 3 |- ( E. y E. z ( ph /\ ps ) <-> ( E. y E. z ph /\ ps ) )
2	1	exbii 1774	. 2 |- ( E. x E. y E. z ( ph /\ ps ) <-> E. x ( E. y E. z ph /\ ps ) )
3		19.41v 1914	. 2 |- ( E. x ( E. y E. z ph /\ ps ) <-> ( E. x E. y E. z ph /\ ps ) )
4	2, 3	bitri 264	1 |- ( E. x E. y E. z ( ph /\ ps ) <-> ( E. x E. y E. z ph /\ 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:  19.41vvvv  1917  eloprabga  6747  dftpos3  7370