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Theorem dfif5 3674
 Description: Alternate definition of the conditional operator df-if 3663. Note that φ is independent of x i.e. a constant true or false (see also abvor0 3567). (Contributed by Gérard Lang, 18-Aug-2013.)
Hypothesis
Ref Expression
dfif3.1 C = {x φ}
Assertion
Ref Expression
dfif5 if(φ, A, B) = ((AB) ∪ (((A B) ∩ C) ∪ ((B A) ∩ (V C))))
Distinct variable group:   φ,x
Allowed substitution hints:   A(x)   B(x)   C(x)

Proof of Theorem dfif5
StepHypRef Expression
1 inindi 3472 . 2 ((AB) ∩ ((A ∪ (V C)) ∩ (BC))) = (((AB) ∩ (A ∪ (V C))) ∩ ((AB) ∩ (BC)))
2 dfif3.1 . . 3 C = {x φ}
32dfif4 3673 . 2 if(φ, A, B) = ((AB) ∩ ((A ∪ (V C)) ∩ (BC)))
4 undir 3504 . . 3 ((AB) ∪ (((A B) ∩ C) ∪ ((B A) ∩ (V C)))) = ((A ∪ (((A B) ∩ C) ∪ ((B A) ∩ (V C)))) ∩ (B ∪ (((A B) ∩ C) ∪ ((B A) ∩ (V C)))))
5 unidm 3407 . . . . . . . 8 (AA) = A
65uneq1i 3414 . . . . . . 7 ((AA) ∪ (B ∩ (V C))) = (A ∪ (B ∩ (V C)))
7 unass 3420 . . . . . . 7 ((AA) ∪ (B ∩ (V C))) = (A ∪ (A ∪ (B ∩ (V C))))
8 undi 3502 . . . . . . 7 (A ∪ (B ∩ (V C))) = ((AB) ∩ (A ∪ (V C)))
96, 7, 83eqtr3ri 2382 . . . . . 6 ((AB) ∩ (A ∪ (V C))) = (A ∪ (A ∪ (B ∩ (V C))))
10 undi 3502 . . . . . . . 8 (A ∪ ((A B) ∩ C)) = ((A ∪ (A B)) ∩ (AC))
11 undifabs 3627 . . . . . . . . . 10 (A ∪ (A B)) = A
1211ineq1i 3453 . . . . . . . . 9 ((A ∪ (A B)) ∩ (AC)) = (A ∩ (AC))
13 inabs 3486 . . . . . . . . 9 (A ∩ (AC)) = A
1412, 13eqtri 2373 . . . . . . . 8 ((A ∪ (A B)) ∩ (AC)) = A
1510, 14eqtri 2373 . . . . . . 7 (A ∪ ((A B) ∩ C)) = A
16 undif2 3626 . . . . . . . . 9 (A ∪ (B A)) = (AB)
1716ineq1i 3453 . . . . . . . 8 ((A ∪ (B A)) ∩ (A ∪ (V C))) = ((AB) ∩ (A ∪ (V C)))
18 undi 3502 . . . . . . . 8 (A ∪ ((B A) ∩ (V C))) = ((A ∪ (B A)) ∩ (A ∪ (V C)))
1917, 18, 83eqtr4i 2383 . . . . . . 7 (A ∪ ((B A) ∩ (V C))) = (A ∪ (B ∩ (V C)))
2015, 19uneq12i 3416 . . . . . 6 ((A ∪ ((A B) ∩ C)) ∪ (A ∪ ((B A) ∩ (V C)))) = (A ∪ (A ∪ (B ∩ (V C))))
219, 20eqtr4i 2376 . . . . 5 ((AB) ∩ (A ∪ (V C))) = ((A ∪ ((A B) ∩ C)) ∪ (A ∪ ((B A) ∩ (V C))))
22 unundi 3424 . . . . 5 (A ∪ (((A B) ∩ C) ∪ ((B A) ∩ (V C)))) = ((A ∪ ((A B) ∩ C)) ∪ (A ∪ ((B A) ∩ (V C))))
2321, 22eqtr4i 2376 . . . 4 ((AB) ∩ (A ∪ (V C))) = (A ∪ (((A B) ∩ C) ∪ ((B A) ∩ (V C))))
24 unass 3420 . . . . . 6 (((AC) ∪ B) ∪ B) = ((AC) ∪ (BB))
25 undi 3502 . . . . . . . . 9 (B ∪ (AC)) = ((BA) ∩ (BC))
26 uncom 3408 . . . . . . . . 9 ((AC) ∪ B) = (B ∪ (AC))
27 undif2 3626 . . . . . . . . . 10 (B ∪ (A B)) = (BA)
2827ineq1i 3453 . . . . . . . . 9 ((B ∪ (A B)) ∩ (BC)) = ((BA) ∩ (BC))
2925, 26, 283eqtr4i 2383 . . . . . . . 8 ((AC) ∪ B) = ((B ∪ (A B)) ∩ (BC))
30 undi 3502 . . . . . . . 8 (B ∪ ((A B) ∩ C)) = ((B ∪ (A B)) ∩ (BC))
3129, 30eqtr4i 2376 . . . . . . 7 ((AC) ∪ B) = (B ∪ ((A B) ∩ C))
32 undi 3502 . . . . . . . 8 (B ∪ ((B A) ∩ (V C))) = ((B ∪ (B A)) ∩ (B ∪ (V C)))
33 undifabs 3627 . . . . . . . . 9 (B ∪ (B A)) = B
3433ineq1i 3453 . . . . . . . 8 ((B ∪ (B A)) ∩ (B ∪ (V C))) = (B ∩ (B ∪ (V C)))
35 inabs 3486 . . . . . . . 8 (B ∩ (B ∪ (V C))) = B
3632, 34, 353eqtrri 2378 . . . . . . 7 B = (B ∪ ((B A) ∩ (V C)))
3731, 36uneq12i 3416 . . . . . 6 (((AC) ∪ B) ∪ B) = ((B ∪ ((A B) ∩ C)) ∪ (B ∪ ((B A) ∩ (V C))))
38 unidm 3407 . . . . . . 7 (BB) = B
3938uneq2i 3415 . . . . . 6 ((AC) ∪ (BB)) = ((AC) ∪ B)
4024, 37, 393eqtr3ri 2382 . . . . 5 ((AC) ∪ B) = ((B ∪ ((A B) ∩ C)) ∪ (B ∪ ((B A) ∩ (V C))))
41 uncom 3408 . . . . . . 7 (BC) = (CB)
4241ineq2i 3454 . . . . . 6 ((AB) ∩ (BC)) = ((AB) ∩ (CB))
43 undir 3504 . . . . . 6 ((AC) ∪ B) = ((AB) ∩ (CB))
4442, 43eqtr4i 2376 . . . . 5 ((AB) ∩ (BC)) = ((AC) ∪ B)
45 unundi 3424 . . . . 5 (B ∪ (((A B) ∩ C) ∪ ((B A) ∩ (V C)))) = ((B ∪ ((A B) ∩ C)) ∪ (B ∪ ((B A) ∩ (V C))))
4640, 44, 453eqtr4i 2383 . . . 4 ((AB) ∩ (BC)) = (B ∪ (((A B) ∩ C) ∪ ((B A) ∩ (V C))))
4723, 46ineq12i 3455 . . 3 (((AB) ∩ (A ∪ (V C))) ∩ ((AB) ∩ (BC))) = ((A ∪ (((A B) ∩ C) ∪ ((B A) ∩ (V C)))) ∩ (B ∪ (((A B) ∩ C) ∪ ((B A) ∩ (V C)))))
484, 47eqtr4i 2376 . 2 ((AB) ∪ (((A B) ∩ C) ∪ ((B A) ∩ (V C)))) = (((AB) ∩ (A ∪ (V C))) ∩ ((AB) ∩ (BC)))
491, 3, 483eqtr4i 2383 1 if(φ, A, B) = ((AB) ∪ (((A B) ∩ C) ∪ ((B A) ∩ (V C))))
 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-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-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: (None)
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