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Theorem nfrab 3478
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.)
Hypotheses
Ref Expression
nfrab.1 𝑥𝜑
nfrab.2 𝑥𝐴
Assertion
Ref Expression
nfrab 𝑥{𝑦𝐴𝜑}

Proof of Theorem nfrab
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-rab 3437 . 2 {𝑦𝐴𝜑} = {𝑦 ∣ (𝑦𝐴𝜑)}
2 nftru 1804 . . . 4 𝑦
3 nfrab.2 . . . . . . . 8 𝑥𝐴
43nfcri 2897 . . . . . . 7 𝑥 𝑧𝐴
5 eleq1w 2824 . . . . . . 7 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
64, 5dvelimnf 2458 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦𝐴)
7 nfrab.1 . . . . . . 7 𝑥𝜑
87a1i 11 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)
96, 8nfand 1897 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑦𝐴𝜑))
109adantl 481 . . . 4 ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜑))
112, 10nfabd2 2929 . . 3 (⊤ → 𝑥{𝑦 ∣ (𝑦𝐴𝜑)})
1211mptru 1547 . 2 𝑥{𝑦 ∣ (𝑦𝐴𝜑)}
131, 12nfcxfr 2903 1 𝑥{𝑦𝐴𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wal 1538  wtru 1541  wnf 1783  wcel 2108  {cab 2714  wnfc 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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