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Theorem dfif4 3673
 Description: Alternate definition of the conditional operator df-if 3663. Note that φ is independent of x i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.)
Hypothesis
Ref Expression
dfif3.1 C = {x φ}
Assertion
Ref Expression
dfif4 if(φ, A, B) = ((AB) ∩ ((A ∪ (V C)) ∩ (BC)))
Distinct variable group:   φ,x
Allowed substitution hints:   A(x)   B(x)   C(x)

Proof of Theorem dfif4
StepHypRef Expression
1 dfif3.1 . . 3 C = {x φ}
21dfif3 3672 . 2 if(φ, A, B) = ((AC) ∪ (B ∩ (V C)))
3 undir 3504 . 2 ((AC) ∪ (B ∩ (V C))) = ((A ∪ (B ∩ (V C))) ∩ (C ∪ (B ∩ (V C))))
4 undi 3502 . . . 4 (A ∪ (B ∩ (V C))) = ((AB) ∩ (A ∪ (V C)))
5 undi 3502 . . . . 5 (C ∪ (B ∩ (V C))) = ((CB) ∩ (C ∪ (V C)))
6 uncom 3408 . . . . . 6 (CB) = (BC)
7 undifv 3624 . . . . . 6 (C ∪ (V C)) = V
86, 7ineq12i 3455 . . . . 5 ((CB) ∩ (C ∪ (V C))) = ((BC) ∩ V)
9 inv1 3577 . . . . 5 ((BC) ∩ V) = (BC)
105, 8, 93eqtri 2377 . . . 4 (C ∪ (B ∩ (V C))) = (BC)
114, 10ineq12i 3455 . . 3 ((A ∪ (B ∩ (V C))) ∩ (C ∪ (B ∩ (V C)))) = (((AB) ∩ (A ∪ (V C))) ∩ (BC))
12 inass 3465 . . 3 (((AB) ∩ (A ∪ (V C))) ∩ (BC)) = ((AB) ∩ ((A ∪ (V C)) ∩ (BC)))
1311, 12eqtri 2373 . 2 ((A ∪ (B ∩ (V C))) ∩ (C ∪ (B ∩ (V C)))) = ((AB) ∩ ((A ∪ (V C)) ∩ (BC)))
142, 3, 133eqtri 2377 1 if(φ, A, B) = ((AB) ∩ ((A ∪ (V C)) ∩ (BC)))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642  {cab 2339  Vcvv 2859   ∖ cdif 3206   ∪ cun 3207   ∩ cin 3208   ifcif 3662 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-ne 2518  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-if 3663 This theorem is referenced by:  dfif5  3674
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