Theorem ifpdc 985

Description: The conditional operator for propositions implies decidability. (Contributed by Jim Kingdon, 25-Jan-2026.)

Assertion

Ref	Expression
ifpdc	|- (if- ( ph , ps , ch ) -> DECID ph )

Proof of Theorem ifpdc

Step	Hyp	Ref	Expression
1		simpl 109	. . 3 |- ( ( ph /\ ps ) -> ph )
2		simpl 109	. . 3 |- ( ( -. ph /\ ch ) -> -. ph )
3	1, 2	orim12i 764	. 2 |- ( ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) -> ( ph \/ -. ph ) )
4		df-ifp 984	. 2 |- (if- ( ph , ps , ch ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) )
5		df-dc 840	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
6	3, 4, 5	3imtr4i 201	1 |- (if- ( ph , ps , ch ) -> DECID ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713  DECID wdc 839  if-wif 983

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-io 714

This theorem depends on definitions:  df-bi 117  df-dc 840  df-ifp 984

This theorem is referenced by:  ifpnst  994