Theorem ifmdc 3645

Description: If a conditional class is inhabited, then the condition is decidable. This shows that conditionals are not very useful unless one can prove the condition decidable. (Contributed by BJ, 24-Sep-2022.)

Assertion

Ref	Expression
ifmdc	|- ( A e. if ( ph , B , C ) -> DECID ph )

Proof of Theorem ifmdc

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2292	. . . . 5 |- ( x = A -> ( x e. if ( ph , B , C ) <-> A e. if ( ph , B , C ) ) )
2	1	imbi1d 231	. . . 4 |- ( x = A -> ( ( x e. if ( ph , B , C ) -> ( ph \/ -. ph ) ) <-> ( A e. if ( ph , B , C ) -> ( ph \/ -. ph ) ) ) )
3		df-if 3603	. . . . . 6 |- if ( ph , B , C ) = { x | ( ( x e. B /\ ph ) \/ ( x e. C /\ -. ph ) ) }
4	3	abeq2i 2340	. . . . 5 |- ( x e. if ( ph , B , C ) <-> ( ( x e. B /\ ph ) \/ ( x e. C /\ -. ph ) ) )
5		simpr 110	. . . . . 6 |- ( ( x e. B /\ ph ) -> ph )
6		simpr 110	. . . . . 6 |- ( ( x e. C /\ -. ph ) -> -. ph )
7	5, 6	orim12i 764	. . . . 5 |- ( ( ( x e. B /\ ph ) \/ ( x e. C /\ -. ph ) ) -> ( ph \/ -. ph ) )
8	4, 7	sylbi 121	. . . 4 |- ( x e. if ( ph , B , C ) -> ( ph \/ -. ph ) )
9	2, 8	vtoclg 2861	. . 3 |- ( A e. if ( ph , B , C ) -> ( A e. if ( ph , B , C ) -> ( ph \/ -. ph ) ) )
10	9	pm2.43i 49	. 2 |- ( A e. if ( ph , B , C ) -> ( ph \/ -. ph ) )
11		df-dc 840	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
12	10, 11	sylibr 134	1 |- ( A e. if ( ph , B , C ) -> DECID ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713  DECID wdc 839   = wceq 1395   e. wcel 2200   ifcif 3602

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  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-dc 840  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-if 3603

This theorem is referenced by: (None)