Theorem 19.42vvvv 1790

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 1788	. . 3 |- ( E. y E. z ( ph /\ ps ) <-> ( ph /\ E. y E. z ps ) )
2	1	2exbii 1497	. 2 |- ( E. w E. x E. y E. z ( ph /\ ps ) <-> E. w E. x ( ph /\ E. y E. z ps ) )
3		19.42vv 1788	. 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 173	1 |- ( E. w E. x E. y E. z ( ph /\ ps ) <-> ( ph /\ E. w E. x E. y E. z ps ) )

Syntax hints:   /\ wa 97   <-> wb 98   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  ceqsex8v  2599  enq0tr  6532