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Mirrors > Home > ILE Home > Th. List > nfifd | GIF version |
Description: Deduction version of nfif 3543. (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 3516 | . 2 ⊢ if(𝜓, 𝐴, 𝐵) = {𝑦 ∣ ((𝑦 ∈ 𝐴 ∧ 𝜓) ∨ (𝑦 ∈ 𝐵 ∧ ¬ 𝜓))} | |
2 | nfv 1515 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfifd.3 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
4 | 3 | nfcrd 2320 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
5 | nfifd.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
6 | 4, 5 | nfand 1555 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓)) |
7 | nfifd.4 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
8 | 7 | nfcrd 2320 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐵) |
9 | 5 | nfnd 1644 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓) |
10 | 8, 9 | nfand 1555 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐵 ∧ ¬ 𝜓)) |
11 | 6, 10 | nford 1554 | . . 3 ⊢ (𝜑 → Ⅎ𝑥((𝑦 ∈ 𝐴 ∧ 𝜓) ∨ (𝑦 ∈ 𝐵 ∧ ¬ 𝜓))) |
12 | 2, 11 | nfabd 2326 | . 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ((𝑦 ∈ 𝐴 ∧ 𝜓) ∨ (𝑦 ∈ 𝐵 ∧ ¬ 𝜓))}) |
13 | 1, 12 | nfcxfrd 2304 | 1 ⊢ (𝜑 → Ⅎ𝑥if(𝜓, 𝐴, 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 698 Ⅎwnf 1447 ∈ wcel 2135 {cab 2150 Ⅎwnfc 2293 ifcif 3515 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-if 3516 |
This theorem is referenced by: nfif 3543 |
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