Theorem ifcldcd 3507

Description: Membership (closure) of a conditional operator, deduction form. (Contributed by Jim Kingdon, 8-Aug-2021.)

Hypotheses

Ref	Expression
ifcldcd.a	|- ( ph -> A e. C )
ifcldcd.b	|- ( ph -> B e. C )
ifcldcd.dc	|- ( ph -> DECID ps )

Assertion

Ref	Expression
ifcldcd	|- ( ph -> if ( ps , A , B ) e. C )

Proof of Theorem ifcldcd

Step	Hyp	Ref	Expression
1		iftrue 3479	. . . 4 |- ( ps -> if ( ps , A , B ) = A )
2	1	adantl 275	. . 3 |- ( ( ph /\ ps ) -> if ( ps , A , B ) = A )
3		ifcldcd.a	. . . 4 |- ( ph -> A e. C )
4	3	adantr 274	. . 3 |- ( ( ph /\ ps ) -> A e. C )
5	2, 4	eqeltrd 2216	. 2 |- ( ( ph /\ ps ) -> if ( ps , A , B ) e. C )
6		iffalse 3482	. . . 4 |- ( -. ps -> if ( ps , A , B ) = B )
7	6	adantl 275	. . 3 |- ( ( ph /\ -. ps ) -> if ( ps , A , B ) = B )
8		ifcldcd.b	. . . 4 |- ( ph -> B e. C )
9	8	adantr 274	. . 3 |- ( ( ph /\ -. ps ) -> B e. C )
10	7, 9	eqeltrd 2216	. 2 |- ( ( ph /\ -. ps ) -> if ( ps , A , B ) e. C )
11		ifcldcd.dc	. . 3 |- ( ph -> DECID ps )
12		df-dc 820	. . 3 |- (DECID ps <-> ( ps \/ -. ps ) )
13	11, 12	sylib 121	. 2 |- ( ph -> ( ps \/ -. ps ) )
14	5, 10, 13	mpjaodan 787	1 |- ( ph -> if ( ps , A , B ) e. C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 697  DECID wdc 819   = wceq 1331   e. wcel 1480   ifcif 3474

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-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-dc 820  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-if 3475

This theorem is referenced by:  fimax2gtrilemstep  6794  fodjuf  7017  fodjum  7018  fodju0  7019  nnnninf  7023  mkvprop  7032  xaddf  9634  xaddval  9635  uzin2  10766  fsum3ser  11173  fsumsplit  11183  explecnv  11281  ennnfonelemp1  11926  nnsf  13209  peano4nninf  13210  nninfalllemn  13212  nninfsellemcl  13217  nninffeq  13226