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| Mirrors > Home > MPE Home > Th. List > nfifd | Structured version Visualization version GIF version | ||
| Description: Deduction form of nfif 4556. (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 4527 | . 2 ⊢ if(𝜓, 𝐴, 𝐵) = {𝑦 ∣ ((𝑦 ∈ 𝐵 → 𝜓) → (𝑦 ∈ 𝐴 ∧ 𝜓))} | |
| 2 | nfv 1914 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
| 3 | nfifd.4 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
| 4 | 3 | nfcrd 2899 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐵) |
| 5 | nfifd.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
| 6 | 4, 5 | nfimd 1894 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐵 → 𝜓)) |
| 7 | nfifd.3 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
| 8 | 7 | nfcrd 2899 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
| 9 | 8, 5 | nfand 1897 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓)) |
| 10 | 6, 9 | nfimd 1894 | . . 3 ⊢ (𝜑 → Ⅎ𝑥((𝑦 ∈ 𝐵 → 𝜓) → (𝑦 ∈ 𝐴 ∧ 𝜓))) |
| 11 | 2, 10 | nfabdw 2927 | . 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ((𝑦 ∈ 𝐵 → 𝜓) → (𝑦 ∈ 𝐴 ∧ 𝜓))}) |
| 12 | 1, 11 | nfcxfrd 2904 | 1 ⊢ (𝜑 → Ⅎ𝑥if(𝜓, 𝐴, 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 Ⅎwnf 1783 ∈ wcel 2108 {cab 2714 Ⅎwnfc 2890 ifcif 4525 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-if 4526 |
| This theorem is referenced by: nfif 4556 nfxnegd 45452 |
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