Theorem 19.41vvv 1897

Description: Theorem 19.41 of [Margaris] p. 90 with 3 quantifiers. (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 1896	. . 3 |- ( E. y E. z ( ph /\ ps ) <-> ( E. y E. z ph /\ ps ) )
2	1	exbii 1598	. 2 |- ( E. x E. y E. z ( ph /\ ps ) <-> E. x ( E. y E. z ph /\ ps ) )
3		19.41v 1895	. 2 |- ( E. x ( E. y E. z ph /\ ps ) <-> ( E. x E. y E. z ph /\ ps ) )
4	2, 3	bitri 183	1 |- ( E. x E. y E. z ( ph /\ ps ) <-> ( E. x E. y E. z ph /\ ps ) )

Syntax hints:   /\ wa 103   <-> wb 104   E.wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  19.41vvvv  1898  eloprabga  5940  dftpos3  6241