Theorem 19.33bdc 1521

Description: Converse of 19.33 1373 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 1520 (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 799	. 2 |- (DECID E. x ph -> ( -. ( E. x ph /\ E. x ps ) <-> ( -. E. x ph \/ -. E. x ps ) ) )
2		19.33b2 1520	. 2 |- ( ( -. E. x ph \/ -. E. x ps ) -> ( A. x ( ph \/ ps ) <-> ( A. x ph \/ A. x ps ) ) )
3	1, 2	syl6bi 152	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 97   <-> wb 98   \/ wo 629  DECID wdc 742   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-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-gen 1338  ax-ie2 1383

This theorem depends on definitions:  df-bi 110  df-dc 743  df-tru 1246  df-fal 1249

This theorem is referenced by: (None)