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Mirrors > Home > MPE Home > Th. List > nfrab | Structured version Visualization version GIF version |
Description: A variable not free in a wff remains so in a restricted class abstraction. Usage of this theorem is discouraged because it depends on ax-13 2366. Use the weaker nfrabw 3467 when possible. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 9-Oct-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfrab.1 | ⊢ Ⅎ𝑥𝜑 |
nfrab.2 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfrab | ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 3431 | . 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} | |
2 | nftru 1798 | . . . 4 ⊢ Ⅎ𝑦⊤ | |
3 | nfrab.2 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
4 | 3 | nfcri 2886 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴 |
5 | eleq1w 2812 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
6 | 4, 5 | dvelimnf 2447 | . . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
7 | nfrab.1 | . . . . . . 7 ⊢ Ⅎ𝑥𝜑 | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑) |
9 | 6, 8 | nfand 1892 | . . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)) |
10 | 9 | adantl 480 | . . . 4 ⊢ ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)) |
11 | 2, 10 | nfabd2 2926 | . . 3 ⊢ (⊤ → Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}) |
12 | 11 | mptru 1540 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} |
13 | 1, 12 | nfcxfr 2897 | 1 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 394 ∀wal 1531 ⊤wtru 1534 Ⅎwnf 1777 ∈ wcel 2098 {cab 2705 Ⅎwnfc 2879 {crab 3430 |
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-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-13 2366 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-rab 3431 |
This theorem is referenced by: elfvmptrab1 7038 elovmporab1 7675 |
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