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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 4889. This shows how the deduction version of a not-free theorem such as nfop 4889 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 3708 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) | |
| 4 | 3 | adantr 480 | . . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) | 
| 5 | abidnf 3708 | . . . 4 ⊢ (Ⅎ𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵) | |
| 6 | 5 | adantl 481 | . . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵) | 
| 7 | 4, 6 | opeq12d 4881 | . 2 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → 〈{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}〉 = 〈𝐴, 𝐵〉) | 
| 8 | nfaba1 2913 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} | |
| 9 | nfaba1 2913 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} | |
| 10 | 8, 9 | nfop 4889 | . 2 ⊢ Ⅎ𝑥〈{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}〉 | 
| 11 | 1, 2, 7, 10 | nfded2 38969 | 1 ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 {cab 2714 Ⅎwnfc 2890 〈cop 4632 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 | 
| This theorem is referenced by: (None) | 
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