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Mirrors > Home > MPE Home > Th. List > nfifd | Structured version Visualization version GIF version |
Description: Deduction form of nfif 4561. (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 4533 | . 2 ⊢ if(𝜓, 𝐴, 𝐵) = {𝑦 ∣ ((𝑦 ∈ 𝐵 → 𝜓) → (𝑦 ∈ 𝐴 ∧ 𝜓))} | |
2 | nfv 1912 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfifd.4 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
4 | 3 | nfcrd 2897 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐵) |
5 | nfifd.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
6 | 4, 5 | nfimd 1892 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐵 → 𝜓)) |
7 | nfifd.3 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
8 | 7 | nfcrd 2897 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
9 | 8, 5 | nfand 1895 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓)) |
10 | 6, 9 | nfimd 1892 | . . 3 ⊢ (𝜑 → Ⅎ𝑥((𝑦 ∈ 𝐵 → 𝜓) → (𝑦 ∈ 𝐴 ∧ 𝜓))) |
11 | 2, 10 | nfabdw 2925 | . 2 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ ((𝑦 ∈ 𝐵 → 𝜓) → (𝑦 ∈ 𝐴 ∧ 𝜓))}) |
12 | 1, 11 | nfcxfrd 2902 | 1 ⊢ (𝜑 → Ⅎ𝑥if(𝜓, 𝐴, 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 Ⅎwnf 1780 ∈ wcel 2106 {cab 2712 Ⅎwnfc 2888 ifcif 4531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-if 4532 |
This theorem is referenced by: nfif 4561 nfxnegd 45391 |
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