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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 2380. Use the weaker nfrabw 3301 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 3077 | . 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} | |
2 | nftru 1807 | . . . 4 ⊢ Ⅎ𝑦⊤ | |
3 | nfrab.2 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
4 | 3 | nfcri 2904 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴 |
5 | eleq1w 2833 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
6 | 4, 5 | dvelimnf 2465 | . . . . . 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 1899 | . . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)) |
10 | 9 | adantl 486 | . . . 4 ⊢ ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)) |
11 | 2, 10 | nfabd2 2940 | . . 3 ⊢ (⊤ → Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}) |
12 | 11 | mptru 1546 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} |
13 | 1, 12 | nfcxfr 2915 | 1 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 400 ∀wal 1537 ⊤wtru 1540 Ⅎwnf 1786 ∈ wcel 2112 {cab 2736 Ⅎwnfc 2897 {crab 3072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-13 2380 ax-ext 2730 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-rab 3077 |
This theorem is referenced by: elfvmptrab1 6779 elovmporab1 7382 |
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