Theorem 19.41vvv 1302

Description: Theorem 19.41 of [Margaris] p. 90 with 3 quantifiers.

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

Proof of Theorem 19.41vvv

Step	Hyp	Ref	Expression
1		19.41vv 1301	. . 3 |- (E.yE.z(ph /\ ps) <-> (E.yE.zph /\ ps))
2	1	exbii 1047	. 2 |- (E.xE.yE.z(ph /\ ps) <-> E.x(E.yE.zph /\ ps))
3		19.41v 1300	. 2 |- (E.x(E.yE.zph /\ ps) <-> (E.xE.yE.zph /\ ps))
4	2, 3	bitr 173	1 |- (E.xE.yE.z(ph /\ ps) <-> (E.xE.yE.zph /\ ps))

Syntax hints:   <-> wb 146   /\ wa 223  E.wex 977

This theorem is referenced by:  eloprabg 3992  19.41vvvv 10335  eeeeanv 10336

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978

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