Theorem 3exbii 1723

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

Hypothesis

Ref	Expression
3exbii.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		3exbii.1	. . 3 |- ( ph <-> ps )
2	1	exbii 1721	. 2 |- ( E. z ph <-> E. z ps )
3	2	2exbii 1722	1 |- ( E. x E. y E. z ph <-> E. x E. y E. z ps )

Syntax hints:   <-> wb 189   E.wex 1666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685

This theorem depends on definitions:  df-bi 190  df-ex 1667

This theorem is referenced by:  4exdistr  1843  ceqsex6v  3057  oprabid  6302  dfoprab2  6324  dftpos3  6977  xpassen  7652  bnj916  29749  bnj917  29750  bnj983  29767  bnj996  29771  bnj1021  29780  bnj1033  29783  ellines  30924