Theorem 19.41vvv 1903

Description: Theorem 19.41 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 30-Apr-1995.)

Assertion

Ref	Expression
19.41vvv	|- (E.xE.yE.z(ph /\ ps) <-> (E.xE.yE.zph /\ 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 1902	. . 3 |- (E.yE.z(ph /\ ps) <-> (E.yE.zph /\ ps))
2	1	exbii 1582	. 2 |- (E.xE.yE.z(ph /\ ps) <-> E.x(E.yE.zph /\ ps))
3		19.41v 1901	. 2 |- (E.x(E.yE.zph /\ ps) <-> (E.xE.yE.zph /\ ps))
4	2, 3	bitri 240	1 |- (E.xE.yE.z(ph /\ ps) <-> (E.xE.yE.zph /\ ps))

This proof depends on syntax axioms:   <-> wb 176   /\ wa 358  E.wex 1541

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

This theorem is used by:  19.41vvvv  1904  eloprabga  5579