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Mirrors > Home > MPE Home > Th. List > nfopd | Structured version Visualization version GIF version |
Description: Deduction version of bound-variable hypothesis builder nfop 4822. This shows how the deduction version of a not-free theorem such as nfop 4822 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 2989 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} | |
2 | nfaba1 2989 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} | |
3 | 1, 2 | nfop 4822 | . 2 ⊢ Ⅎ𝑥〈{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}〉 |
4 | nfopd.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | nfopd.3 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
6 | nfnfc1 2983 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | |
7 | nfnfc1 2983 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐵 | |
8 | 6, 7 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) |
9 | abidnf 3697 | . . . . . 6 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) | |
10 | 9 | adantr 483 | . . . . 5 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) |
11 | abidnf 3697 | . . . . . 6 ⊢ (Ⅎ𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵) | |
12 | 11 | adantl 484 | . . . . 5 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵) |
13 | 10, 12 | opeq12d 4814 | . . . 4 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → 〈{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}〉 = 〈𝐴, 𝐵〉) |
14 | 8, 13 | nfceqdf 2975 | . . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → (Ⅎ𝑥〈{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}〉 ↔ Ⅎ𝑥〈𝐴, 𝐵〉)) |
15 | 4, 5, 14 | syl2anc 586 | . 2 ⊢ (𝜑 → (Ⅎ𝑥〈{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}, {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}〉 ↔ Ⅎ𝑥〈𝐴, 𝐵〉)) |
16 | 3, 15 | mpbii 235 | 1 ⊢ (𝜑 → Ⅎ𝑥〈𝐴, 𝐵〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1534 = wceq 1536 ∈ wcel 2113 {cab 2802 Ⅎwnfc 2964 〈cop 4576 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 |
This theorem is referenced by: nfbrd 5115 dfid3 5465 nfovd 7188 |
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