Theorem 3exbii 1543

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 1541	. 2 |- ( E. z ph <-> E. z ps )
3	2	2exbii 1542	1 |- ( E. x E. y E. z ph <-> E. x E. y E. z ps )

Syntax hints:   <-> wb 103   E.wex 1426

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  eeeanv  1856  ceqsex6v  2663  oprabid  5681  dfoprab2  5696  dftpos3  6027  xpassen  6546