Theorem 19.30dc 1627

Description: Theorem 19.30 of [Margaris] p. 90, with an additional decidability condition. (Contributed by Jim Kingdon, 21-Jul-2018.)

Assertion

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

Proof of Theorem 19.30dc

Step	Hyp	Ref	Expression
1		df-dc 835	. 2 |- (DECID E. x ps <-> ( E. x ps \/ -. E. x ps ) )
2		olc 711	. . . 4 |- ( E. x ps -> ( A. x ph \/ E. x ps ) )
3	2	a1d 22	. . 3 |- ( E. x ps -> ( A. x ( ph \/ ps ) -> ( A. x ph \/ E. x ps ) ) )
4		alnex 1499	. . . . 5 |- ( A. x -. ps <-> -. E. x ps )
5		orel2 726	. . . . . 6 |- ( -. ps -> ( ( ph \/ ps ) -> ph ) )
6	5	al2imi 1458	. . . . 5 |- ( A. x -. ps -> ( A. x ( ph \/ ps ) -> A. x ph ) )
7	4, 6	sylbir 135	. . . 4 |- ( -. E. x ps -> ( A. x ( ph \/ ps ) -> A. x ph ) )
8		orc 712	. . . 4 |- ( A. x ph -> ( A. x ph \/ E. x ps ) )
9	7, 8	syl6 33	. . 3 |- ( -. E. x ps -> ( A. x ( ph \/ ps ) -> ( A. x ph \/ E. x ps ) ) )
10	3, 9	jaoi 716	. 2 |- ( ( E. x ps \/ -. E. x ps ) -> ( A. x ( ph \/ ps ) -> ( A. x ph \/ E. x ps ) ) )
11	1, 10	sylbi 121	1 |- (DECID E. x ps -> ( A. x ( ph \/ ps ) -> ( A. x ph \/ E. x ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 708  DECID wdc 834   A.wal 1351   E.wex 1492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-gen 1449  ax-ie2 1494

This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-fal 1359

This theorem is referenced by: (None)