Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2386. (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 2452 | . . . . 5 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦 | |
3 | 1, 2 | nfan 1896 | . . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) |
4 | nfabd2.2 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) | |
5 | 3, 4 | nfabd 3001 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
6 | 5 | ex 415 | . 2 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥{𝑦 ∣ 𝜓})) |
7 | nfab1 2979 | . . 3 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜓} | |
8 | eqidd 2822 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑦 ∣ 𝜓} = {𝑦 ∣ 𝜓}) | |
9 | 8 | drnfc1 2997 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥{𝑦 ∣ 𝜓} ↔ Ⅎ𝑦{𝑦 ∣ 𝜓})) |
10 | 7, 9 | mpbiri 260 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
11 | 6, 10 | pm2.61d2 183 | 1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∀wal 1531 Ⅎwnf 1780 {cab 2799 Ⅎwnfc 2961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-13 2386 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 |
This theorem is referenced by: nfabdOLD 3004 nfrab 3386 nfixp 8480 |
Copyright terms: Public domain | W3C validator |