Theorem pm10.57 38570

Description: Theorem *10.57 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
pm10.57	|- ( A. x ( ph -> ( ps \/ ch ) ) -> ( A. x ( ph -> ps ) \/ E. x ( ph /\ ch ) ) )

Proof of Theorem pm10.57

Step	Hyp	Ref	Expression
1		alnex 1706	. . . 4 |- ( A. x -. ( ph /\ ch ) <-> -. E. x ( ph /\ ch ) )
2		imnan 438	. . . . . 6 |- ( ( ph -> -. ch ) <-> -. ( ph /\ ch ) )
3		pm2.53 388	. . . . . . . 8 |- ( ( ps \/ ch ) -> ( -. ps -> ch ) )
4	3	con1d 139	. . . . . . 7 |- ( ( ps \/ ch ) -> ( -. ch -> ps ) )
5	4	imim3i 64	. . . . . 6 |- ( ( ph -> ( ps \/ ch ) ) -> ( ( ph -> -. ch ) -> ( ph -> ps ) ) )
6	2, 5	syl5bir 233	. . . . 5 |- ( ( ph -> ( ps \/ ch ) ) -> ( -. ( ph /\ ch ) -> ( ph -> ps ) ) )
7	6	al2imi 1743	. . . 4 |- ( A. x ( ph -> ( ps \/ ch ) ) -> ( A. x -. ( ph /\ ch ) -> A. x ( ph -> ps ) ) )
8	1, 7	syl5bir 233	. . 3 |- ( A. x ( ph -> ( ps \/ ch ) ) -> ( -. E. x ( ph /\ ch ) -> A. x ( ph -> ps ) ) )
9	8	con1d 139	. 2 |- ( A. x ( ph -> ( ps \/ ch ) ) -> ( -. A. x ( ph -> ps ) -> E. x ( ph /\ ch ) ) )
10	9	orrd 393	1 |- ( A. x ( ph -> ( ps \/ ch ) ) -> ( A. x ( ph -> ps ) \/ E. x ( ph /\ ch ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   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-an 386  df-ex 1705

This theorem is referenced by: (None)