Theorem 19.36vv 38582

Description: Theorem *11.43 in [WhiteheadRussell] p. 163. Theorem 19.36 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
19.36vv	|- ( E. x E. y ( ph -> ps ) <-> ( A. x A. y ph -> ps ) )

Distinct variable groups:   ps, x   ps, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem 19.36vv

Step	Hyp	Ref	Expression
1		19.36v 1904	. . 3 |- ( E. y ( ph -> ps ) <-> ( A. y ph -> ps ) )
2	1	exbii 1774	. 2 |- ( E. x E. y ( ph -> ps ) <-> E. x ( A. y ph -> ps ) )
3		19.36v 1904	. 2 |- ( E. x ( A. y ph -> ps ) <-> ( A. x A. y ph -> ps ) )
4	2, 3	bitri 264	1 |- ( E. x E. y ( ph -> ps ) <-> ( A. x A. y ph -> 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  ax-5 1839  ax-6 1888

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

This theorem is referenced by: (None)