Theorem 3exbii 1776

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 1774	. 2 |- ( E. z ph <-> E. z ps )
3	2	2exbii 1775	1 |- ( E. x E. y E. z ph <-> E. x E. y E. z ps )

Syntax hints:   <-> wb 196   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  4exdistr  1924  ceqsex6v  3248  oprabid  6677  dfoprab2  6701  dftpos3  7370  xpassen  8054  bnj916  31003  bnj917  31004  bnj983  31021  bnj996  31025  bnj1021  31034  bnj1033  31037  ellines  32259