Theorem ifcl 2380

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 1534	. 2 |- (A = if(ph, A, B) -> (A e. C <-> if(ph, A, B) e. C))
2		eleq1 1534	. 2 |- (B = if(ph, A, B) -> (B e. C <-> if(ph, A, B) e. C))
3	1, 2	ifboth 2375	1 |- ((A e. C /\ B e. C) -> if(ph, A, B) e. C)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 958  ifcif 2361

This theorem is referenced by:  ifpr 2427  suppr 4590  xrmaxltt 5913  xrltmint 5914  maxlet 5918  lemint 5921  maxltt 5922  z2get 6188  iooint 6372  fsequb 6523  seq1bnd 6910  caubnd 6926  clm3 7079  ivthlem7 7287  retopbas 7655  xpcn 7976  iscms2lem4 7992  spwval2 8653

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-if 2362

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