Theorem 19.42vvvv 1885

Description: Theorem 19.42 of [Margaris] p. 90 with 4 quantifiers. (Contributed by Jim Kingdon, 23-Nov-2019.)

Assertion

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

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

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

Proof of Theorem 19.42vvvv

Step	Hyp	Ref	Expression
1		19.42vv 1883	. . 3 |- ( E. y E. z ( ph /\ ps ) <-> ( ph /\ E. y E. z ps ) )
2	1	2exbii 1585	. 2 |- ( E. w E. x E. y E. z ( ph /\ ps ) <-> E. w E. x ( ph /\ E. y E. z ps ) )
3		19.42vv 1883	. 2 |- ( E. w E. x ( ph /\ E. y E. z ps ) <-> ( ph /\ E. w E. x E. y E. z ps ) )
4	2, 3	bitri 183	1 |- ( E. w E. x E. y E. z ( ph /\ ps ) <-> ( ph /\ E. w E. x E. y E. z ps ) )

Syntax hints:   /\ wa 103   <-> wb 104   E.wex 1468

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  ceqsex8v  2726  enq0tr  7235