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Mirrors > Home > MPE Home > Th. List > Mathboxes > nfopdALT | Structured version Visualization version GIF version |
Description: Deduction version of bound-variable hypothesis builder nfop 4449. This shows how the deduction version of a not-free theorem such as nfop 4449 can be created from the corresponding not-free inference theorem. (Contributed by NM, 19-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfopdALT.1 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfopdALT.2 | ⊢ (𝜑 → Ⅎ𝑥𝐵) |
Ref | Expression |
---|---|
nfopdALT | ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfopdALT.1 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
2 | nfopdALT.2 | . 2 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
3 | abidnf 3408 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) | |
4 | 3 | adantr 480 | . . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) |
5 | abidnf 3408 | . . . 4 ⊢ (Ⅎ𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵) | |
6 | 5 | adantl 481 | . . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵) |
7 | 4, 6 | opeq12d 4441 | . 2 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → 〈{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}〉 = 〈𝐴, 𝐵〉) |
8 | nfaba1 2799 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} | |
9 | nfaba1 2799 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} | |
10 | 8, 9 | nfop 4449 | . 2 ⊢ Ⅎ𝑥〈{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}〉 |
11 | 1, 2, 7, 10 | nfded2 34573 | 1 ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1521 = wceq 1523 ∈ wcel 2030 {cab 2637 Ⅎwnfc 2780 〈cop 4216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 |
This theorem is referenced by: (None) |
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