![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > nfifd | GIF version |
Description: Deduction version of nfif 3577. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Ref | Expression |
---|---|
nfifd.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
nfifd.3 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfifd.4 | ⊢ (𝜑 → Ⅎ𝑥𝐵) |
Ref | Expression |
---|---|
nfifd | ⊢ (𝜑 → Ⅎ𝑥if(𝜓, 𝐴, 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-if 3550 | . 2 ⊢ if(𝜓, 𝐴, 𝐵) = {𝑦 ∣ ((𝑦 ∈ 𝐴 ∧ 𝜓) ∨ (𝑦 ∈ 𝐵 ∧ ¬ 𝜓))} | |
2 | nfv 1539 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfifd.3 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
4 | 3 | nfcrd 2346 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
5 | nfifd.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
6 | 4, 5 | nfand 1579 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓)) |
7 | nfifd.4 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
8 | 7 | nfcrd 2346 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐵) |
9 | 5 | nfnd 1668 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓) |
10 | 8, 9 | nfand 1579 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐵 ∧ ¬ 𝜓)) |
11 | 6, 10 | nford 1578 | . . 3 ⊢ (𝜑 → Ⅎ𝑥((𝑦 ∈ 𝐴 ∧ 𝜓) ∨ (𝑦 ∈ 𝐵 ∧ ¬ 𝜓))) |
12 | 2, 11 | nfabd 2352 | . 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ((𝑦 ∈ 𝐴 ∧ 𝜓) ∨ (𝑦 ∈ 𝐵 ∧ ¬ 𝜓))}) |
13 | 1, 12 | nfcxfrd 2330 | 1 ⊢ (𝜑 → Ⅎ𝑥if(𝜓, 𝐴, 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 709 Ⅎwnf 1471 ∈ wcel 2160 {cab 2175 Ⅎwnfc 2319 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-in1 615 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-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 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-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-if 3550 |
This theorem is referenced by: nfif 3577 |
Copyright terms: Public domain | W3C validator |