Theorem 2exbidv 1796

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 1753	. 2 |- ( ph -> ( E. y ps <-> E. y ch ) )
3	2	exbidv 1753	1 |- ( ph -> ( E. x E. y ps <-> E. x E. y ch ) )

Syntax hints:   -> wi 4   <-> 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-17 1464  ax-ial 1472

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  3exbidv  1797  4exbidv  1798  cbvex4v  1853  ceqsex3v  2661  ceqsex4v  2662  copsexg  4071  euotd  4081  elopab  4085  elxpi  4454  relop  4586  cbvoprab3  5724  ov6g  5782  th3qlem1  6392  ltresr  7374  fisumcom2  10828