Theorem ifcl 2370

Description: Membership (closure) of a conditional operator.

Assertion

Ref	Expression
ifcl	|- ((A e. C /\ B e. C) -> if(ph, A, B) e. C)

Proof of Theorem ifcl

Step	Hyp	Ref	Expression
1		eleq1 1526	. 2 |- (A = if(ph, A, B) -> (A e. C <-> if(ph, A, B) e. C))
2		eleq1 1526	. 2 |- (B = if(ph, A, B) -> (B e. C <-> if(ph, A, B) e. C))
3	1, 2	ifboth 2365	1 |- ((A e. C /\ B e. C) -> if(ph, A, B) e. C)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 955  ifcif 2351

This theorem is referenced by:  ifpr 2417  suppr 4562  xrmaxltt 5861  xrltmint 5862  maxlet 5866  lemint 5869  maxltt 5870  z2get 6135  iooint 6309  fsequb 6455  seq1bnd 6847  caubnd 6863  clm3 7017  ivthlem7 7222  ivthlem7OLD 7231  retopbas 7597  xpcn 7910  iscms2lem4 7926  spwval2 8577

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-if 2352

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