Theorem ifcldcd 3454

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 3426	. . . 4 |- ( ps -> if ( ps , A , B ) = A )
2	1	adantl 273	. . 3 |- ( ( ph /\ ps ) -> if ( ps , A , B ) = A )
3		ifcldcd.a	. . . 4 |- ( ph -> A e. C )
4	3	adantr 272	. . 3 |- ( ( ph /\ ps ) -> A e. C )
5	2, 4	eqeltrd 2176	. 2 |- ( ( ph /\ ps ) -> if ( ps , A , B ) e. C )
6		iffalse 3429	. . . 4 |- ( -. ps -> if ( ps , A , B ) = B )
7	6	adantl 273	. . 3 |- ( ( ph /\ -. ps ) -> if ( ps , A , B ) = B )
8		ifcldcd.b	. . . 4 |- ( ph -> B e. C )
9	8	adantr 272	. . 3 |- ( ( ph /\ -. ps ) -> B e. C )
10	7, 9	eqeltrd 2176	. 2 |- ( ( ph /\ -. ps ) -> if ( ps , A , B ) e. C )
11		ifcldcd.dc	. . 3 |- ( ph -> DECID ps )
12		df-dc 787	. . 3 |- (DECID ps <-> ( ps \/ -. ps ) )
13	11, 12	sylib 121	. 2 |- ( ph -> ( ps \/ -. ps ) )
14	5, 10, 13	mpjaodan 753	1 |- ( ph -> if ( ps , A , B ) e. C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 670  DECID wdc 786   = wceq 1299   e. wcel 1448   ifcif 3421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-11 1452  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-dc 787  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-if 3422

This theorem is referenced by:  fimax2gtrilemstep  6723  fodjuf  6929  fodjum  6930  fodju0  6931  nnnninf  6935  mkvprop  6943  xaddf  9468  xaddval  9469  uzin2  10599  fsum3ser  11005  fsumsplit  11015  explecnv  11113  ennnfonelemp1  11711  nnsf  12783  peano4nninf  12784  nninfalllemn  12786  nninfsellemcl  12791  nninffeq  12800