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Mirrors > Home > NFE Home > Th. List > dfif4 | GIF version |
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.) |
Ref | Expression |
---|---|
dfif3.1 | ⊢ C = {x ∣ φ} |
Ref | Expression |
---|---|
dfif4 | ⊢ if(φ, A, B) = ((A ∪ B) ∩ ((A ∪ (V ∖ C)) ∩ (B ∪ C))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfif3.1 | . . 3 ⊢ C = {x ∣ φ} | |
2 | 1 | dfif3 3672 | . 2 ⊢ if(φ, A, B) = ((A ∩ C) ∪ (B ∩ (V ∖ C))) |
3 | undir 3504 | . 2 ⊢ ((A ∩ C) ∪ (B ∩ (V ∖ C))) = ((A ∪ (B ∩ (V ∖ C))) ∩ (C ∪ (B ∩ (V ∖ C)))) | |
4 | undi 3502 | . . . 4 ⊢ (A ∪ (B ∩ (V ∖ C))) = ((A ∪ B) ∩ (A ∪ (V ∖ C))) | |
5 | undi 3502 | . . . . 5 ⊢ (C ∪ (B ∩ (V ∖ C))) = ((C ∪ B) ∩ (C ∪ (V ∖ C))) | |
6 | uncom 3408 | . . . . . 6 ⊢ (C ∪ B) = (B ∪ C) | |
7 | undifv 3624 | . . . . . 6 ⊢ (C ∪ (V ∖ C)) = V | |
8 | 6, 7 | ineq12i 3455 | . . . . 5 ⊢ ((C ∪ B) ∩ (C ∪ (V ∖ C))) = ((B ∪ C) ∩ V) |
9 | inv1 3577 | . . . . 5 ⊢ ((B ∪ C) ∩ V) = (B ∪ C) | |
10 | 5, 8, 9 | 3eqtri 2377 | . . . 4 ⊢ (C ∪ (B ∩ (V ∖ C))) = (B ∪ C) |
11 | 4, 10 | ineq12i 3455 | . . 3 ⊢ ((A ∪ (B ∩ (V ∖ C))) ∩ (C ∪ (B ∩ (V ∖ C)))) = (((A ∪ B) ∩ (A ∪ (V ∖ C))) ∩ (B ∪ C)) |
12 | inass 3465 | . . 3 ⊢ (((A ∪ B) ∩ (A ∪ (V ∖ C))) ∩ (B ∪ C)) = ((A ∪ B) ∩ ((A ∪ (V ∖ C)) ∩ (B ∪ C))) | |
13 | 11, 12 | eqtri 2373 | . 2 ⊢ ((A ∪ (B ∩ (V ∖ C))) ∩ (C ∪ (B ∩ (V ∖ C)))) = ((A ∪ B) ∩ ((A ∪ (V ∖ C)) ∩ (B ∪ C))) |
14 | 2, 3, 13 | 3eqtri 2377 | 1 ⊢ if(φ, A, B) = ((A ∪ B) ∩ ((A ∪ (V ∖ C)) ∩ (B ∪ 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: dfif5 3674 |
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