Theorem 19.30 1809

Description: Theorem 19.30 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
19.30	|- ( A. x ( ph \/ ps ) -> ( A. x ph \/ E. x ps ) )

Proof of Theorem 19.30

Step	Hyp	Ref	Expression
1		exnal 1754	. . 3 |- ( E. x -. ph <-> -. A. x ph )
2		pm2.53 388	. . . 4 |- ( ( ph \/ ps ) -> ( -. ph -> ps ) )
3	2	aleximi 1759	. . 3 |- ( A. x ( ph \/ ps ) -> ( E. x -. ph -> E. x ps ) )
4	1, 3	syl5bir 233	. 2 |- ( A. x ( ph \/ ps ) -> ( -. A. x ph -> E. x ps ) )
5	4	orrd 393	1 |- ( A. x ( ph \/ ps ) -> ( A. x ph \/ E. x ps ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   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-or 385  df-ex 1705

This theorem is referenced by:  19.33b  1813