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 4582  xrmaxltt 5901  xrltmint 5902  maxlet 5906  lemint 5909  maxltt 5910  z2get 6176  iooint 6358  fsequb 6509  seq1bnd 6895  caubnd 6911  clm3 7064  ivthlem7 7272  retopbas 7640  xpcn 7961  iscms2lem4 7977  spwval2 8638

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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