Theorem 2exbidv 1891

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 1848	. 2 |- ( ph -> ( E. y ps <-> E. y ch ) )
3	2	exbidv 1848	1 |- ( ph -> ( E. x E. y ps <-> E. x E. y ch ) )

Syntax hints:   -> wi 4   <-> wb 105   E.wex 1515

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  3exbidv  1892  4exbidv  1893  cbvex4v  1958  ceqsex3v  2815  ceqsex4v  2816  copsexg  4288  euotd  4299  elopab  4304  elxpi  4691  relop  4828  cbvoprab3  6021  ov6g  6084  th3qlem1  6724  ltresr  7952  fisumcom2  11749  fprodcom2fi  11937