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Mirrors > Home > MPE Home > Th. List > Mathboxes > nfded | Structured version Visualization version GIF version |
Description: A deduction theorem that converts a not-free inference directly to deduction form. The first hypothesis is the hypothesis of the deduction form. The second is an equality deduction (e.g., (Ⅎ𝑥𝐴 → ∪ {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴} = ∪ 𝐴)) that starts from abidnf 3691. The last is assigned to the inference form (e.g., Ⅎ𝑥∪ {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}) whose hypothesis is satisfied using nfaba1 2983. (Contributed by NM, 19-Nov-2020.) |
Ref | Expression |
---|---|
nfded.1 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfded.2 | ⊢ (Ⅎ𝑥𝐴 → 𝐵 = 𝐶) |
nfded.3 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nfded | ⊢ (𝜑 → Ⅎ𝑥𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfded.3 | . 2 ⊢ Ⅎ𝑥𝐵 | |
2 | nfded.1 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
3 | nfnfc1 2977 | . . . 4 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | |
4 | nfded.2 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → 𝐵 = 𝐶) | |
5 | 3, 4 | nfceqdf 2969 | . . 3 ⊢ (Ⅎ𝑥𝐴 → (Ⅎ𝑥𝐵 ↔ Ⅎ𝑥𝐶)) |
6 | 2, 5 | syl 17 | . 2 ⊢ (𝜑 → (Ⅎ𝑥𝐵 ↔ Ⅎ𝑥𝐶)) |
7 | 1, 6 | mpbii 234 | 1 ⊢ (𝜑 → Ⅎ𝑥𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 Ⅎwnfc 2958 |
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-cleq 2811 df-clel 2890 df-nfc 2960 |
This theorem is referenced by: nfunidALT 35986 |
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