Theorem 2exbi 38579

Description: Theorem *11.341 in [WhiteheadRussell] p. 162. Theorem 19.18 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
2exbi	|- ( A. x A. y ( ph <-> ps ) -> ( E. x E. y ph <-> E. x E. y ps ) )

Proof of Theorem 2exbi

Step	Hyp	Ref	Expression
1		exbi 1773	. . 3 |- ( A. y ( ph <-> ps ) -> ( E. y ph <-> E. y ps ) )
2	1	alimi 1739	. 2 |- ( A. x A. y ( ph <-> ps ) -> A. x ( E. y ph <-> E. y ps ) )
3		exbi 1773	. 2 |- ( A. x ( E. y ph <-> E. y ps ) -> ( E. x E. y ph <-> E. x E. y ps ) )
4	2, 3	syl 17	1 |- ( A. x A. y ( ph <-> ps ) -> ( E. x E. y ph <-> E. x E. y ps ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by: (None)