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Mirrors > Home > ILE Home > Th. List > nfopd | GIF version |
Description: Deduction version of bound-variable hypothesis builder nfop 3781. This shows how the deduction version of a not-free theorem such as nfop 3781 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.) |
Ref | Expression |
---|---|
nfopd.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfopd.3 | ⊢ (𝜑 → Ⅎ𝑥𝐵) |
Ref | Expression |
---|---|
nfopd | ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfaba1 2318 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} | |
2 | nfaba1 2318 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} | |
3 | 1, 2 | nfop 3781 | . 2 ⊢ Ⅎ𝑥〈{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}〉 |
4 | nfopd.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | nfopd.3 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
6 | nfnfc1 2315 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | |
7 | nfnfc1 2315 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐵 | |
8 | 6, 7 | nfan 1558 | . . . 4 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) |
9 | abidnf 2898 | . . . . . 6 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) | |
10 | 9 | adantr 274 | . . . . 5 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) |
11 | abidnf 2898 | . . . . . 6 ⊢ (Ⅎ𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵) | |
12 | 11 | adantl 275 | . . . . 5 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵) |
13 | 10, 12 | opeq12d 3773 | . . . 4 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → 〈{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}〉 = 〈𝐴, 𝐵〉) |
14 | 8, 13 | nfceqdf 2311 | . . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → (Ⅎ𝑥〈{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}〉 ↔ Ⅎ𝑥〈𝐴, 𝐵〉)) |
15 | 4, 5, 14 | syl2anc 409 | . 2 ⊢ (𝜑 → (Ⅎ𝑥〈{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}〉 ↔ Ⅎ𝑥〈𝐴, 𝐵〉)) |
16 | 3, 15 | mpbii 147 | 1 ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1346 = wceq 1348 ∈ wcel 2141 {cab 2156 Ⅎwnfc 2299 〈cop 3586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-sn 3589 df-pr 3590 df-op 3592 |
This theorem is referenced by: nfbrd 4034 nfovd 5882 |
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