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Theorem nfrab 3377
 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 2392. Use the weaker nfrabw 3376 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 3142 . 2 {𝑦𝐴𝜑} = {𝑦 ∣ (𝑦𝐴𝜑)}
2 nftru 1806 . . . 4 𝑦
3 nfrab.2 . . . . . . . 8 𝑥𝐴
43nfcri 2969 . . . . . . 7 𝑥 𝑧𝐴
5 eleq1w 2898 . . . . . . 7 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
64, 5dvelimnf 2477 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦𝐴)
7 nfrab.1 . . . . . . 7 𝑥𝜑
87a1i 11 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)
96, 8nfand 1899 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑦𝐴𝜑))
109adantl 485 . . . 4 ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜑))
112, 10nfabd2 3005 . . 3 (⊤ → 𝑥{𝑦 ∣ (𝑦𝐴𝜑)})
1211mptru 1545 . 2 𝑥{𝑦 ∣ (𝑦𝐴𝜑)}
131, 12nfcxfr 2980 1 𝑥{𝑦𝐴𝜑}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 399  ∀wal 1536  ⊤wtru 1539  Ⅎwnf 1785   ∈ wcel 2115  {cab 2802  Ⅎwnfc 2962  {crab 3137 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-13 2392  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142 This theorem is referenced by:  elfvmptrab1  6786  elovmporab1  7387
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