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Theorem nfrab 3471
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 2366. Use the weaker nfrabw 3467 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 3431 . 2 {𝑦𝐴𝜑} = {𝑦 ∣ (𝑦𝐴𝜑)}
2 nftru 1798 . . . 4 𝑦
3 nfrab.2 . . . . . . . 8 𝑥𝐴
43nfcri 2886 . . . . . . 7 𝑥 𝑧𝐴
5 eleq1w 2812 . . . . . . 7 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
64, 5dvelimnf 2447 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦𝐴)
7 nfrab.1 . . . . . . 7 𝑥𝜑
87a1i 11 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)
96, 8nfand 1892 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑦𝐴𝜑))
109adantl 480 . . . 4 ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜑))
112, 10nfabd2 2926 . . 3 (⊤ → 𝑥{𝑦 ∣ (𝑦𝐴𝜑)})
1211mptru 1540 . 2 𝑥{𝑦 ∣ (𝑦𝐴𝜑)}
131, 12nfcxfr 2897 1 𝑥{𝑦𝐴𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 394  wal 1531  wtru 1534  wnf 1777  wcel 2098  {cab 2705  wnfc 2879  {crab 3430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-13 2366  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-rab 3431
This theorem is referenced by:  elfvmptrab1  7038  elovmporab1  7675
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