Theorem 19.41vvv 1904

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 1903	. . 3 |- ( E. y E. z ( ph /\ ps ) <-> ( E. y E. z ph /\ ps ) )
2	1	exbii 1605	. 2 |- ( E. x E. y E. z ( ph /\ ps ) <-> E. x ( E. y E. z ph /\ ps ) )
3		19.41v 1902	. 2 |- ( E. x ( E. y E. z ph /\ ps ) <-> ( E. x E. y E. z ph /\ ps ) )
4	2, 3	bitri 184	1 |- ( E. x E. y E. z ( ph /\ ps ) <-> ( E. x E. y E. z ph /\ ps ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1492

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.41vvvv  1905  eloprabga  5964  dftpos3  6265