Theorem 2exbidv 1890

Description: Formula-building rule for 2 existential quantifiers (deduction form). (Contributed by NM, 1-May-1995.)

Hypothesis

Ref	Expression
2albidv.1	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
2exbidv	|- ( ph -> ( E. x E. y ps <-> E. x E. y ch ) )

Distinct variable groups:   ph, x   ph, y

Allowed substitution hints:   ps( x, y)   ch( x, y)

Proof of Theorem 2exbidv

Step	Hyp	Ref	Expression
1		2albidv.1	. . 3 |- ( ph -> ( ps <-> ch ) )
2	1	exbidv 1847	. 2 |- ( ph -> ( E. y ps <-> E. y ch ) )
3	2	exbidv 1847	1 |- ( ph -> ( E. x E. y ps <-> E. x E. y ch ) )

Syntax hints:   -> wi 4   <-> wb 105   E.wex 1514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-17 1548  ax-ial 1556

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  3exbidv  1891  4exbidv  1892  cbvex4v  1957  ceqsex3v  2814  ceqsex4v  2815  copsexg  4287  euotd  4298  elopab  4303  elxpi  4690  relop  4827  cbvoprab3  6020  ov6g  6083  th3qlem1  6723  ltresr  7951  fisumcom2  11720  fprodcom2fi  11908