Theorem 2exbidv 1914

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 1871	. 2 |- ( ph -> ( E. y ps <-> E. y ch ) )
3	2	exbidv 1871	1 |- ( ph -> ( E. x E. y ps <-> E. x E. y ch ) )

Syntax hints:   -> wi 4   <-> wb 105   E.wex 1538

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  3exbidv  1915  4exbidv  1916  cbvex4v  1981  ceqsex3v  2843  ceqsex4v  2844  copsexg  4330  euotd  4341  elopab  4346  elxpi  4735  relop  4872  cbvoprab3  6080  ov6g  6143  th3qlem1  6784  ltresr  8026  fisumcom2  11949  fprodcom2fi  12137