Theorem 2exbidv 1882

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 1839	. 2 |- ( ph -> ( E. y ps <-> E. y ch ) )
3	2	exbidv 1839	1 |- ( ph -> ( E. x E. y ps <-> E. x E. y ch ) )

Syntax hints:   -> wi 4   <-> wb 105   E.wex 1506

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  3exbidv  1883  4exbidv  1884  cbvex4v  1949  ceqsex3v  2806  ceqsex4v  2807  copsexg  4278  euotd  4288  elopab  4293  elxpi  4680  relop  4817  cbvoprab3  6002  ov6g  6065  th3qlem1  6705  ltresr  7923  fisumcom2  11620  fprodcom2fi  11808