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Mirrors > Home > ILE Home > Th. List > ifsbdc | GIF version |
Description: Distribute a function over an if-clause. (Contributed by Jim Kingdon, 1-Jan-2022.) |
Ref | Expression |
---|---|
ifsbdc.1 | ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → 𝐶 = 𝐷) |
ifsbdc.2 | ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → 𝐶 = 𝐸) |
Ref | Expression |
---|---|
ifsbdc | ⊢ (DECID 𝜑 → 𝐶 = if(𝜑, 𝐷, 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmiddc 821 | . 2 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑)) | |
2 | iftrue 3474 | . . . . 5 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
3 | ifsbdc.1 | . . . . 5 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → 𝐶 = 𝐷) | |
4 | 2, 3 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐶 = 𝐷) |
5 | iftrue 3474 | . . . 4 ⊢ (𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐷) | |
6 | 4, 5 | eqtr4d 2173 | . . 3 ⊢ (𝜑 → 𝐶 = if(𝜑, 𝐷, 𝐸)) |
7 | iffalse 3477 | . . . . 5 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵) | |
8 | ifsbdc.2 | . . . . 5 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → 𝐶 = 𝐸) | |
9 | 7, 8 | syl 14 | . . . 4 ⊢ (¬ 𝜑 → 𝐶 = 𝐸) |
10 | iffalse 3477 | . . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐸) | |
11 | 9, 10 | eqtr4d 2173 | . . 3 ⊢ (¬ 𝜑 → 𝐶 = if(𝜑, 𝐷, 𝐸)) |
12 | 6, 11 | jaoi 705 | . 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → 𝐶 = if(𝜑, 𝐷, 𝐸)) |
13 | 1, 12 | syl 14 | 1 ⊢ (DECID 𝜑 → 𝐶 = if(𝜑, 𝐷, 𝐸)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 697 DECID wdc 819 = wceq 1331 ifcif 3469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-if 3470 |
This theorem is referenced by: fvifdc 5436 |
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