Theorem 3exbidv 1862

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

Hypothesis

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

Assertion

Ref	Expression
3exbidv	|- ( ph -> ( E. x E. y E. z ps <-> E. x E. y E. z ch ) )

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 1818	. 2 |- ( ph -> ( E. z ps <-> E. z ch ) )
3	2	2exbidv 1861	1 |- ( ph -> ( E. x E. y E. z ps <-> E. x E. y E. z ch ) )

Syntax hints:   -> wi 4   <-> wb 104   E.wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  ceqsex6v  2774  euotd  4239  oprabid  5885  eloprabga  5940  eloprabi  6175