Theorem 2exbidv 1748

Description: Formula-building rule for 2 existential quantifiers (deduction rule). (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 1706	. 2 |- ( ph -> ( E. y ps <-> E. y ch ) )
3	2	exbidv 1706	1 |- ( ph -> ( E. x E. y ps <-> E. x E. y ch ) )

Syntax hints:   -> wi 4   <-> wb 98   E.wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  3exbidv  1749  4exbidv  1750  cbvex4v  1805  ceqsex3v  2596  ceqsex4v  2597  copsexg  3981  euotd  3991  elopab  3995  elxpi  4361  relop  4486  cbvoprab3  5580  ov6g  5638  th3qlem1  6208  ltresr  6915