Theorem 2exbidv 1921

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 1878	. 2 |- ( ph -> ( E. y ps <-> E. y ch ) )
3	2	exbidv 1878	1 |- ( ph -> ( E. x E. y ps <-> E. x E. y ch ) )

This proof depends on syntax axioms:   -> wi 4   <-> 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-17 1579  ax-ial 1587

This proof depends on definitions:  df-bi 117

This theorem is used by:  3exbidv  1922  4exbidv  1923  cbvex4v  1990  ceqsex3v  2865  ceqsex4v  2866  copsexg  4384  euotd  4395  elopab  4400  elxpi  4790  relop  4930  cbvoprab3  6164  ov6g  6227  th3qlem1  6911  ltresr  8206  fisumcom2  12205  fprodcom2fi  12393