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Theorem ifcldcd 3358
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
StepHypRef Expression
1 iftrue 3336 . . . 4  |-  ( ps 
->  if ( ps ,  A ,  B )  =  A )
21adantl 262 . . 3  |-  ( (
ph  /\  ps )  ->  if ( ps ,  A ,  B )  =  A )
3 ifcldcd.a . . . 4  |-  ( ph  ->  A  e.  C )
43adantr 261 . . 3  |-  ( (
ph  /\  ps )  ->  A  e.  C )
52, 4eqeltrd 2114 . 2  |-  ( (
ph  /\  ps )  ->  if ( ps ,  A ,  B )  e.  C )
6 iffalse 3339 . . . 4  |-  ( -. 
ps  ->  if ( ps ,  A ,  B
)  =  B )
76adantl 262 . . 3  |-  ( (
ph  /\  -.  ps )  ->  if ( ps ,  A ,  B )  =  B )
8 ifcldcd.b . . . 4  |-  ( ph  ->  B  e.  C )
98adantr 261 . . 3  |-  ( (
ph  /\  -.  ps )  ->  B  e.  C )
107, 9eqeltrd 2114 . 2  |-  ( (
ph  /\  -.  ps )  ->  if ( ps ,  A ,  B )  e.  C )
11 ifcldcd.dc . . 3  |-  ( ph  -> DECID  ps )
12 df-dc 743 . . 3  |-  (DECID  ps  <->  ( ps  \/  -.  ps ) )
1311, 12sylib 127 . 2  |-  ( ph  ->  ( ps  \/  -.  ps ) )
145, 10, 13mpjaodan 711 1  |-  ( ph  ->  if ( ps ,  A ,  B )  e.  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 97    \/ wo 629  DECID wdc 742    = wceq 1243    e. wcel 1393   ifcif 3331
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-dc 743  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-if 3332
This theorem is referenced by:  uzin2  9586
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