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Mirrors > Home > MPE Home > Th. List > nfabd2 | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2363. (Contributed by Mario Carneiro, 8-Oct-2016.) (Proof shortened by Wolf Lammen, 10-May-2023.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfabd2.1 | ⊢ Ⅎ𝑦𝜑 |
nfabd2.2 | ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfabd2 | ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfabd2.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
2 | nfnae 2425 | . . . . 5 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦 | |
3 | 1, 2 | nfan 1894 | . . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) |
4 | nfabd2.2 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) | |
5 | 3, 4 | nfabd 2920 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
6 | 5 | ex 412 | . 2 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥{𝑦 ∣ 𝜓})) |
7 | nfab1 2897 | . . 3 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜓} | |
8 | eqidd 2725 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑦 ∣ 𝜓} = {𝑦 ∣ 𝜓}) | |
9 | 8 | drnfc1 2914 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥{𝑦 ∣ 𝜓} ↔ Ⅎ𝑦{𝑦 ∣ 𝜓})) |
10 | 7, 9 | mpbiri 258 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
11 | 6, 10 | pm2.61d2 181 | 1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1531 Ⅎwnf 1777 {cab 2701 Ⅎwnfc 2875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-13 2363 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-nfc 2877 |
This theorem is referenced by: nfrab 3464 nfixp 8908 |
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