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Mirrors > Home > MPE Home > Th. List > nfabd2OLD | Structured version Visualization version GIF version |
Description: Obsolete version of nfabd2 3004 as of 23-May-2023. (Contributed by Mario Carneiro, 8-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfabd2.1 | ⊢ Ⅎ𝑦𝜑 |
nfabd2.2 | ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfabd2OLD | ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑧(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) | |
2 | df-clab 2802 | . . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓) | |
3 | nfabd2.1 | . . . . . . 7 ⊢ Ⅎ𝑦𝜑 | |
4 | nfnae 2456 | . . . . . . 7 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦 | |
5 | 3, 4 | nfan 1900 | . . . . . 6 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) |
6 | nfabd2.2 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) | |
7 | 5, 6 | nfsbd 2564 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥[𝑧 / 𝑦]𝜓) |
8 | 2, 7 | nfxfrd 1854 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜓}) |
9 | 1, 8 | nfcd 2970 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
10 | 9 | ex 415 | . 2 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥{𝑦 ∣ 𝜓})) |
11 | nfab1 2981 | . . 3 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜓} | |
12 | eqidd 2824 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑦 ∣ 𝜓} = {𝑦 ∣ 𝜓}) | |
13 | 12 | drnfc1 2999 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥{𝑦 ∣ 𝜓} ↔ Ⅎ𝑦{𝑦 ∣ 𝜓})) |
14 | 11, 13 | mpbiri 260 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
15 | 10, 14 | pm2.61d2 183 | 1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∀wal 1535 Ⅎwnf 1784 [wsb 2069 ∈ wcel 2114 {cab 2801 Ⅎwnfc 2963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 |
This theorem is referenced by: (None) |
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