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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 837 | . 2 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑)) | |
2 | iftrue 3554 | . . . . 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 3554 | . . . 4 ⊢ (𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐷) | |
6 | 4, 5 | eqtr4d 2225 | . . 3 ⊢ (𝜑 → 𝐶 = if(𝜑, 𝐷, 𝐸)) |
7 | iffalse 3557 | . . . . 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 3557 | . . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐸) | |
11 | 9, 10 | eqtr4d 2225 | . . 3 ⊢ (¬ 𝜑 → 𝐶 = if(𝜑, 𝐷, 𝐸)) |
12 | 6, 11 | jaoi 717 | . 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 709 DECID wdc 835 = wceq 1364 ifcif 3549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-11 1517 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-if 3550 |
This theorem is referenced by: fvifdc 5556 lgsneg 14883 lgsdilem 14886 |
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