Theorem 3exbidv 1629

Description: Formula-building rule for 3 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)

Hypothesis

Ref	Expression
3exbidv.1	|- (ph -> (ps <-> ch))

Assertion

Ref	Expression
3exbidv	|- (ph -> (E.xE.yE.zps <-> E.xE.yE.zch))

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

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

Proof of Theorem 3exbidv

Step	Hyp	Ref	Expression
1		3exbidv.1	. . 3 |- (ph -> (ps <-> ch))
2	1	exbidv 1626	. 2 |- (ph -> (E.zps <-> E.zch))
3	2	2exbidv 1628	1 |- (ph -> (E.xE.yE.zps <-> E.xE.yE.zch))

Syntax hints:   -> wi 4   <-> wb 176  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

This theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by:  ceqsex6v  2900  ins2keq  4219  ins3keq  4220  opkelins2kg  4252  opkelins3kg  4253  oprabid  5551  eloprabga  5579