New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  dfif3 GIF version

Theorem dfif3 3672
 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.) (Revised by Mario Carneiro, 8-Sep-2013.)
Hypothesis
Ref Expression
dfif3.1 C = {x φ}
Assertion
Ref Expression
dfif3 if(φ, A, B) = ((AC) ∪ (B ∩ (V C)))
Distinct variable group:   φ,x
Allowed substitution hints:   A(x)   B(x)   C(x)

Proof of Theorem dfif3
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfif6 3665 . 2 if(φ, A, B) = ({y A φ} ∪ {y B ¬ φ})
2 dfif3.1 . . . . . 6 C = {x φ}
3 biidd 228 . . . . . . 7 (x = y → (φφ))
43cbvabv 2472 . . . . . 6 {x φ} = {y φ}
52, 4eqtri 2373 . . . . 5 C = {y φ}
65ineq2i 3454 . . . 4 (AC) = (A ∩ {y φ})
7 dfrab3 3531 . . . 4 {y A φ} = (A ∩ {y φ})
86, 7eqtr4i 2376 . . 3 (AC) = {y A φ}
9 dfrab3 3531 . . . 4 {y B ¬ φ} = (B ∩ {y ¬ φ})
10 notab 3525 . . . . . 6 {y ¬ φ} = (V {y φ})
115difeq2i 3382 . . . . . 6 (V C) = (V {y φ})
1210, 11eqtr4i 2376 . . . . 5 {y ¬ φ} = (V C)
1312ineq2i 3454 . . . 4 (B ∩ {y ¬ φ}) = (B ∩ (V C))
149, 13eqtr2i 2374 . . 3 (B ∩ (V C)) = {y B ¬ φ}
158, 14uneq12i 3416 . 2 ((AC) ∪ (B ∩ (V C))) = ({y A φ} ∪ {y B ¬ φ})
161, 15eqtr4i 2376 1 if(φ, A, B) = ((AC) ∪ (B ∩ (V C)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1642  {cab 2339  {crab 2618  Vcvv 2859   ∖ cdif 3206   ∪ cun 3207   ∩ cin 3208   ifcif 3662 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-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-if 3663 This theorem is referenced by:  dfif4  3673
 Copyright terms: Public domain W3C validator