Theorem 19.33bdc 1610

Description: Converse of 19.33 1461 given -. ( E. x ph /\ E. x ps ) and a decidability condition. Compare Theorem 19.33 of [Margaris] p. 90. For a version which does not require a decidability condition, see 19.33b2 1609 (Contributed by Jim Kingdon, 23-Apr-2018.)

Assertion

Ref	Expression
19.33bdc	|- (DECID E. x ph -> ( -. ( E. x ph /\ E. x ps ) -> ( A. x ( ph \/ ps ) <-> ( A. x ph \/ A. x ps ) ) ) )

Proof of Theorem 19.33bdc

Step	Hyp	Ref	Expression
1		ianordc 885	. 2 |- (DECID E. x ph -> ( -. ( E. x ph /\ E. x ps ) <-> ( -. E. x ph \/ -. E. x ps ) ) )
2		19.33b2 1609	. 2 |- ( ( -. E. x ph \/ -. E. x ps ) -> ( A. x ( ph \/ ps ) <-> ( A. x ph \/ A. x ps ) ) )
3	1, 2	syl6bi 162	1 |- (DECID E. x ph -> ( -. ( E. x ph /\ E. x ps ) -> ( A. x ( ph \/ ps ) <-> ( A. x ph \/ A. x ps ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  DECID wdc 820   A.wal 1330   E.wex 1469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-gen 1426  ax-ie2 1471

This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1335  df-fal 1338

This theorem is referenced by: (None)