Theorem 19.23vv 1764

Description: Theorem 19.23 of [Margaris] p. 90 extended to two variables. (Contributed by NM, 10-Aug-2004.)

Assertion

Ref	Expression
19.23vv	|- ( A. x A. y ( ph -> ps ) <-> ( E. x E. y ph -> ps ) )

Distinct variable groups:   ps, x   ps, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem 19.23vv

Step	Hyp	Ref	Expression
1		19.23v 1763	. . 3 |- ( A. y ( ph -> ps ) <-> ( E. y ph -> ps ) )
2	1	albii 1359	. 2 |- ( A. x A. y ( ph -> ps ) <-> A. x ( E. y ph -> ps ) )
3		19.23v 1763	. 2 |- ( A. x ( E. y ph -> ps ) <-> ( E. x E. y ph -> ps ) )
4	2, 3	bitri 173	1 |- ( A. x A. y ( ph -> ps ) <-> ( E. x E. y ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   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-ie2 1383  ax-17 1419

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  ssrel  4428  ssrelrel  4440  raliunxp  4477