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Mirrors > Home > MPE Home > Th. List > Mathboxes > nfded2 | Structured version Visualization version GIF version |
Description: A deduction theorem that converts a not-free inference directly to deduction form. The first 2 hypotheses are the hypotheses of the deduction form. The third is an equality deduction (e.g., ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → 〈{𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}, {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐵}〉 = 〈𝐴, 𝐵〉) for nfopd 4787) that starts from abidnf 3605. The last is assigned to the inference form (e.g., Ⅎ𝑥〈{𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}, {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐵}〉 for nfop 4786) whose hypotheses are satisfied using nfaba1 2905. (Contributed by NM, 19-Nov-2020.) |
Ref | Expression |
---|---|
nfded2.1 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfded2.2 | ⊢ (𝜑 → Ⅎ𝑥𝐵) |
nfded2.3 | ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → 𝐶 = 𝐷) |
nfded2.4 | ⊢ Ⅎ𝑥𝐶 |
Ref | Expression |
---|---|
nfded2 | ⊢ (𝜑 → Ⅎ𝑥𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfded2.4 | . 2 ⊢ Ⅎ𝑥𝐶 | |
2 | nfded2.1 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
3 | nfded2.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
4 | nfnfc1 2900 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | |
5 | nfnfc1 2900 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐵 | |
6 | 4, 5 | nfan 1907 | . . . 4 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) |
7 | nfded2.3 | . . . 4 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → 𝐶 = 𝐷) | |
8 | 6, 7 | nfceqdf 2892 | . . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → (Ⅎ𝑥𝐶 ↔ Ⅎ𝑥𝐷)) |
9 | 2, 3, 8 | syl2anc 587 | . 2 ⊢ (𝜑 → (Ⅎ𝑥𝐶 ↔ Ⅎ𝑥𝐷)) |
10 | 1, 9 | mpbii 236 | 1 ⊢ (𝜑 → Ⅎ𝑥𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 Ⅎwnfc 2877 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-cleq 2728 df-nfc 2879 |
This theorem is referenced by: nfopdALT 36671 |
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