Theorem 19.42vvv 1908

Description: Theorem 19.42 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 21-Sep-2011.)

Assertion

Ref	Expression
19.42vvv	|- (E.xE.yE.z(ph /\ ps) <-> (ph /\ E.xE.yE.zps))

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

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

Proof of Theorem 19.42vvv

Step	Hyp	Ref	Expression
1		19.42vv 1907	. . 3 |- (E.yE.z(ph /\ ps) <-> (ph /\ E.yE.zps))
2	1	exbii 1582	. 2 |- (E.xE.yE.z(ph /\ ps) <-> E.x(ph /\ E.yE.zps))
3		19.42v 1905	. 2 |- (E.x(ph /\ E.yE.zps) <-> (ph /\ E.xE.yE.zps))
4	2, 3	bitri 240	1 |- (E.xE.yE.z(ph /\ ps) <-> (ph /\ E.xE.yE.zps))

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545

This theorem is referenced by:  ceqsex6v  2900