Theorem 3exbidv 1749

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. 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 1706	. 2 |- ( ph -> ( E. z ps <-> E. z ch ) )
3	2	2exbidv 1748	1 |- ( ph -> ( E. x E. y E. z ps <-> E. x E. y E. z ch ) )

Syntax hints:   -> wi 4   <-> wb 98   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  ceqsex6v  2598  euotd  3991  oprabid  5537  eloprabga  5591  eloprabi  5822