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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 4852) that starts from abidnf 3665. The last is assigned to the inference form (e.g., Ⅎ𝑥⟨{𝑦 ∣ ∀𝑥𝑦 ∈ 𝐴}, {𝑦 ∣ ∀𝑥𝑦 ∈ 𝐵}⟩ for nfop 4851) whose hypotheses are satisfied using nfaba1 2916. (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 2911 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | |
5 | nfnfc1 2911 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐵 | |
6 | 4, 5 | nfan 1903 | . . . 4 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) |
7 | nfded2.3 | . . . 4 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → 𝐶 = 𝐷) | |
8 | 6, 7 | nfceqdf 2903 | . . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → (Ⅎ𝑥𝐶 ↔ Ⅎ𝑥𝐷)) |
9 | 2, 3, 8 | syl2anc 585 | . 2 ⊢ (𝜑 → (Ⅎ𝑥𝐶 ↔ Ⅎ𝑥𝐷)) |
10 | 1, 9 | mpbii 232 | 1 ⊢ (𝜑 → Ⅎ𝑥𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 Ⅎwnfc 2888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-cleq 2729 df-nfc 2890 |
This theorem is referenced by: nfopdALT 37462 |
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