Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > nfopdALT | Structured version Visualization version GIF version |
Description: Deduction version of bound-variable hypothesis builder nfop 4418. This shows how the deduction version of a not-free theorem such as nfop 4418 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 3375 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) | |
4 | 3 | adantr 481 | . . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) |
5 | abidnf 3375 | . . . 4 ⊢ (Ⅎ𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵) | |
6 | 5 | adantl 482 | . . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵) |
7 | 4, 6 | opeq12d 4410 | . 2 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → 〈{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}〉 = 〈𝐴, 𝐵〉) |
8 | nfaba1 2770 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} | |
9 | nfaba1 2770 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} | |
10 | 8, 9 | nfop 4418 | . 2 ⊢ Ⅎ𝑥〈{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}〉 |
11 | 1, 2, 7, 10 | nfded2 34255 | 1 ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∀wal 1481 = wceq 1483 ∈ wcel 1990 {cab 2608 Ⅎwnfc 2751 〈cop 4183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |