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Mirrors > Home > ILE Home > Th. List > dedhb | GIF version |
Description: A deduction theorem for converting the inference ⊢ Ⅎ𝑥𝐴 => ⊢ 𝜑 into a closed theorem. Use nfa1 1502 and nfab 2258 to eliminate the hypothesis of the substitution instance 𝜓 of the inference. For converting the inference form into a deduction form, abidnf 2819 is useful. (Contributed by NM, 8-Dec-2006.) |
Ref | Expression |
---|---|
dedhb.1 | ⊢ (𝐴 = {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} → (𝜑 ↔ 𝜓)) |
dedhb.2 | ⊢ 𝜓 |
Ref | Expression |
---|---|
dedhb | ⊢ (Ⅎ𝑥𝐴 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dedhb.2 | . 2 ⊢ 𝜓 | |
2 | abidnf 2819 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) | |
3 | 2 | eqcomd 2118 | . . 3 ⊢ (Ⅎ𝑥𝐴 → 𝐴 = {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}) |
4 | dedhb.1 | . . 3 ⊢ (𝐴 = {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} → (𝜑 ↔ 𝜓)) | |
5 | 3, 4 | syl 14 | . 2 ⊢ (Ⅎ𝑥𝐴 → (𝜑 ↔ 𝜓)) |
6 | 1, 5 | mpbiri 167 | 1 ⊢ (Ⅎ𝑥𝐴 → 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1310 = wceq 1312 ∈ wcel 1461 {cab 2099 Ⅎwnfc 2240 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-11 1465 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 |
This theorem is referenced by: (None) |
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