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| Description: Deduction version of bound-variable hypothesis builder nfop 4888. This shows how the deduction version of a not-free theorem such as nfop 4888 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 2912 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} | |
| 2 | nfaba1 2912 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} | |
| 3 | 1, 2 | nfop 4888 | . 2 ⊢ Ⅎ𝑥〈{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}〉 | 
| 4 | nfopd.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
| 5 | nfopd.3 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
| 6 | nfnfc1 2907 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | |
| 7 | nfnfc1 2907 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐵 | |
| 8 | 6, 7 | nfan 1898 | . . . 4 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) | 
| 9 | abidnf 3707 | . . . . . 6 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) | |
| 10 | 9 | adantr 480 | . . . . 5 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) | 
| 11 | abidnf 3707 | . . . . . 6 ⊢ (Ⅎ𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵) | |
| 12 | 11 | adantl 481 | . . . . 5 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵) | 
| 13 | 10, 12 | opeq12d 4880 | . . . 4 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → 〈{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}〉 = 〈𝐴, 𝐵〉) | 
| 14 | 8, 13 | nfceqdf 2900 | . . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → (Ⅎ𝑥〈{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}〉 ↔ Ⅎ𝑥〈𝐴, 𝐵〉)) | 
| 15 | 4, 5, 14 | syl2anc 584 | . 2 ⊢ (𝜑 → (Ⅎ𝑥〈{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}〉 ↔ Ⅎ𝑥〈𝐴, 𝐵〉)) | 
| 16 | 3, 15 | mpbii 233 | 1 ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1537 = wceq 1539 ∈ wcel 2107 {cab 2713 Ⅎwnfc 2889 〈cop 4631 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 | 
| This theorem is referenced by: nfbrd 5188 dfid3 5580 nfovd 7461 | 
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