Theorem 3exbii 1586

Description: Inference adding 3 existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.)

Hypothesis

Ref	Expression
exbii.1	|- ( ph <-> ps )

Assertion

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

Proof of Theorem 3exbii

Step	Hyp	Ref	Expression
1		exbii.1	. . 3 |- ( ph <-> ps )
2	1	exbii 1584	. 2 |- ( E. z ph <-> E. z ps )
3	2	2exbii 1585	1 |- ( E. x E. y E. z ph <-> E. x E. y E. z ps )

Syntax hints:   <-> wb 104   E.wex 1468

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  eeeanv  1905  ceqsex6v  2730  oprabid  5803  dfoprab2  5818  dftpos3  6159  xpassen  6724