Theorem 3exbii 1660

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 1658	. 2 |- ( E. z ph <-> E. z ps )
3	2	2exbii 1659	1 |- ( E. x E. y E. z ph <-> E. x E. y E. z ps )

This proof depends on syntax axioms:   <-> wb 105   E.wex 1545

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-ial 1587

This proof depends on definitions:  df-bi 117

This theorem is used by:  eeeanv  1993  ceqsex6v  2867  oprabid  6117  dfoprab2  6135  dftpos3  6533  xpassen  7128