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Theorem ifpr 3774
 Description: Membership of a conditional operator in an unordered pair. (Contributed by NM, 17-Jun-2007.)
Assertion
Ref Expression
ifpr ((A C B D) → if(φ, A, B) {A, B})

Proof of Theorem ifpr
StepHypRef Expression
1 elex 2867 . 2 (A CA V)
2 elex 2867 . 2 (B DB V)
3 ifcl 3698 . . 3 ((A V B V) → if(φ, A, B) V)
4 ifeqor 3699 . . . 4 ( if(φ, A, B) = A if(φ, A, B) = B)
5 elprg 3750 . . . 4 ( if(φ, A, B) V → ( if(φ, A, B) {A, B} ↔ ( if(φ, A, B) = A if(φ, A, B) = B)))
64, 5mpbiri 224 . . 3 ( if(φ, A, B) V → if(φ, A, B) {A, B})
73, 6syl 15 . 2 ((A V B V) → if(φ, A, B) {A, B})
81, 2, 7syl2an 463 1 ((A C B D) → if(φ, A, B) {A, B})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ifcif 3662  {cpr 3738 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-if 3663  df-sn 3741  df-pr 3742 This theorem is referenced by: (None)
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