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Mirrors > Home > MPE Home > Th. List > nfifd | Structured version Visualization version GIF version |
Description: Deduction form of nfif 4554. (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 4526 | . 2 ⊢ if(𝜓, 𝐴, 𝐵) = {𝑦 ∣ ((𝑦 ∈ 𝐵 → 𝜓) → (𝑦 ∈ 𝐴 ∧ 𝜓))} | |
2 | nfv 1910 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfifd.4 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
4 | 3 | nfcrd 2887 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐵) |
5 | nfifd.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
6 | 4, 5 | nfimd 1890 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐵 → 𝜓)) |
7 | nfifd.3 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
8 | 7 | nfcrd 2887 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
9 | 8, 5 | nfand 1893 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓)) |
10 | 6, 9 | nfimd 1890 | . . 3 ⊢ (𝜑 → Ⅎ𝑥((𝑦 ∈ 𝐵 → 𝜓) → (𝑦 ∈ 𝐴 ∧ 𝜓))) |
11 | 2, 10 | nfabdw 2921 | . 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ((𝑦 ∈ 𝐵 → 𝜓) → (𝑦 ∈ 𝐴 ∧ 𝜓))}) |
12 | 1, 11 | nfcxfrd 2897 | 1 ⊢ (𝜑 → Ⅎ𝑥if(𝜓, 𝐴, 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 Ⅎwnf 1778 ∈ wcel 2099 {cab 2704 Ⅎwnfc 2878 ifcif 4524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-if 4525 |
This theorem is referenced by: nfif 4554 nfxnegd 44795 |
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