Theorem anifpdc 992

Description: The conditional operator is implied by the conjunction of its possible outputs. Dual statement of ifpor 993. (Contributed by BJ, 30-Sep-2019.)

Assertion

Ref	Expression
anifpdc	|- (DECID ph -> ( ( ps /\ ch ) -> if- ( ph , ps , ch ) ) )

Proof of Theorem anifpdc

Step	Hyp	Ref	Expression
1		olc 716	. . 3 |- ( ps -> ( -. ph \/ ps ) )
2		olc 716	. . 3 |- ( ch -> ( ph \/ ch ) )
3	1, 2	anim12i 338	. 2 |- ( ( ps /\ ch ) -> ( ( -. ph \/ ps ) /\ ( ph \/ ch ) ) )
4		dfifp4dc 989	. 2 |- (DECID ph -> (if- ( ph , ps , ch ) <-> ( ( -. ph \/ ps ) /\ ( ph \/ ch ) ) ) )
5	3, 4	imbitrrid 156	1 |- (DECID ph -> ( ( ps /\ ch ) -> if- ( ph , ps , ch ) ) )

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-in1 617  ax-in2 618  ax-io 714

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

This theorem is referenced by: (None)