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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. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 9-Oct-2016.) |
Ref | Expression |
---|---|
nfrab.1 | ⊢ Ⅎ𝑥𝜑 |
nfrab.2 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfrab | ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 2905 | . 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} | |
2 | nftru 1721 | . . . 4 ⊢ Ⅎ𝑦⊤ | |
3 | nfrab.2 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
4 | 3 | nfcri 2745 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴 |
5 | eleq1 2676 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
6 | 4, 5 | dvelimnf 2327 | . . . . . 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 1814 | . . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)) |
10 | 9 | adantl 481 | . . . 4 ⊢ ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)) |
11 | 2, 10 | nfabd2 2770 | . . 3 ⊢ (⊤ → Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}) |
12 | 11 | trud 1484 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} |
13 | 1, 12 | nfcxfr 2749 | 1 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 ∀wal 1473 ⊤wtru 1476 Ⅎwnf 1699 ∈ wcel 1977 {cab 2596 Ⅎwnfc 2738 {crab 2900 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 |
This theorem is referenced by: nfdif 3693 nfin 3782 nfse 5013 elfvmptrab1 6213 elovmpt2rab 6778 elovmpt2rab1 6779 ovmpt3rab1 6789 elovmpt3rab1 6791 mpt2xopoveq 7232 nfoi 8302 scottex 8631 elmptrab 21440 iundisjf 28784 nnindf 28952 bnj1398 30356 bnj1445 30366 bnj1449 30370 nfwlim 31012 finminlem 31482 poimirlem26 32605 poimirlem27 32606 indexa 32698 binomcxplemdvbinom 37574 binomcxplemdvsum 37576 binomcxplemnotnn0 37577 fnlimfvre 38741 fnlimabslt 38746 dvnprodlem1 38836 stoweidlem16 38909 stoweidlem31 38924 stoweidlem34 38927 stoweidlem35 38928 stoweidlem48 38941 stoweidlem51 38944 stoweidlem53 38946 stoweidlem54 38947 stoweidlem57 38950 stoweidlem59 38952 fourierdlem31 39031 fourierdlem48 39047 fourierdlem51 39050 etransclem32 39159 ovncvrrp 39454 smflim 39663 |
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