Theorem 19.42vvv 1927

Description: Theorem 19.42 of [Margaris] p. 90 with 3 quantifiers. (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 1926	. . 3 |- ( E. y E. z ( ph /\ ps ) <-> ( ph /\ E. y E. z ps ) )
2	1	exbii 1619	. 2 |- ( E. x E. y E. z ( ph /\ ps ) <-> E. x ( ph /\ E. y E. z ps ) )
3		19.42v 1921	. 2 |- ( E. x ( ph /\ E. y E. z ps ) <-> ( ph /\ E. x E. y E. z ps ) )
4	2, 3	bitri 184	1 |- ( E. x E. y E. z ( ph /\ ps ) <-> ( ph /\ E. x E. y E. z ps ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  ceqsex6v  2808