Theorem ifcldcd 3502

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 3474	. . . 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 2214	. 2 |- ( ( ph /\ ps ) -> if ( ps , A , B ) e. C )
6		iffalse 3477	. . . 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 2214	. 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 3469

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 2119

This theorem depends on definitions:  df-bi 116  df-dc 820  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-if 3470

This theorem is referenced by:  fimax2gtrilemstep  6787  fodjuf  7010  fodjum  7011  fodju0  7012  nnnninf  7016  mkvprop  7025  xaddf  9620  xaddval  9621  uzin2  10752  fsum3ser  11159  fsumsplit  11169  explecnv  11267  ennnfonelemp1  11908  nnsf  13188  peano4nninf  13189  nninfalllemn  13191  nninfsellemcl  13196  nninffeq  13205