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| 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 2377. Use the weaker nfrabw 3475 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 3437 | . 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} | |
| 2 | nftru 1804 | . . . 4 ⊢ Ⅎ𝑦⊤ | |
| 3 | nfrab.2 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
| 4 | 3 | nfcri 2897 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴 | 
| 5 | eleq1w 2824 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
| 6 | 4, 5 | dvelimnf 2458 | . . . . . 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 1897 | . . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)) | 
| 10 | 9 | adantl 481 | . . . 4 ⊢ ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)) | 
| 11 | 2, 10 | nfabd2 2929 | . . 3 ⊢ (⊤ → Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}) | 
| 12 | 11 | mptru 1547 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} | 
| 13 | 1, 12 | nfcxfr 2903 | 1 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝜑} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ∧ wa 395 ∀wal 1538 ⊤wtru 1541 Ⅎwnf 1783 ∈ wcel 2108 {cab 2714 Ⅎwnfc 2890 {crab 3436 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rab 3437 | 
| This theorem is referenced by: elfvmptrab1 7044 elovmporab1 7681 | 
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