Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nfifd | Structured version Visualization version GIF version |
Description: Deduction form of nfif 4492. (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 | dfif2 4465 | . 2 ⊢ if(𝜓, 𝐴, 𝐵) = {𝑦 ∣ ((𝑦 ∈ 𝐵 → 𝜓) → (𝑦 ∈ 𝐴 ∧ 𝜓))} | |
2 | nfv 1906 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfifd.4 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
4 | 3 | nfcrd 2966 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐵) |
5 | nfifd.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
6 | 4, 5 | nfimd 1886 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐵 → 𝜓)) |
7 | nfifd.3 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
8 | 7 | nfcrd 2966 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
9 | 8, 5 | nfand 1889 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓)) |
10 | 6, 9 | nfimd 1886 | . . 3 ⊢ (𝜑 → Ⅎ𝑥((𝑦 ∈ 𝐵 → 𝜓) → (𝑦 ∈ 𝐴 ∧ 𝜓))) |
11 | 2, 10 | nfabdw 2997 | . 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ((𝑦 ∈ 𝐵 → 𝜓) → (𝑦 ∈ 𝐴 ∧ 𝜓))}) |
12 | 1, 11 | nfcxfrd 2973 | 1 ⊢ (𝜑 → Ⅎ𝑥if(𝜓, 𝐴, 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 Ⅎwnf 1775 ∈ wcel 2105 {cab 2796 Ⅎwnfc 2958 ifcif 4463 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-if 4464 |
This theorem is referenced by: nfif 4492 nfxnegd 41591 |
Copyright terms: Public domain | W3C validator |