Theorem 19.41vvvv 1785

Description: Theorem 19.41 of [Margaris] p. 90 with 4 quantifiers. (Contributed by FL, 14-Jul-2007.)

Assertion

Ref	Expression
19.41vvvv	|- ( E. w E. x E. y E. z ( ph /\ ps ) <-> ( E. w E. x E. y E. z ph /\ ps ) )

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

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

Proof of Theorem 19.41vvvv

Step	Hyp	Ref	Expression
1		19.41vvv 1784	. . 3 |- ( E. x E. y E. z ( ph /\ ps ) <-> ( E. x E. y E. z ph /\ ps ) )
2	1	exbii 1496	. 2 |- ( E. w E. x E. y E. z ( ph /\ ps ) <-> E. w ( E. x E. y E. z ph /\ ps ) )
3		19.41v 1782	. 2 |- ( E. w ( E. x E. y E. z ph /\ ps ) <-> ( E. w E. x E. y E. z ph /\ ps ) )
4	2, 3	bitri 173	1 |- ( E. w E. x E. y E. z ( ph /\ ps ) <-> ( E. w E. x E. y E. z ph /\ 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: (None)