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Theorem nfrab 3476
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 2375. Use the weaker nfrabw 3473 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 3434 . 2 {𝑦𝐴𝜑} = {𝑦 ∣ (𝑦𝐴𝜑)}
2 nftru 1801 . . . 4 𝑦
3 nfrab.2 . . . . . . . 8 𝑥𝐴
43nfcri 2895 . . . . . . 7 𝑥 𝑧𝐴
5 eleq1w 2822 . . . . . . 7 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
64, 5dvelimnf 2456 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦𝐴)
7 nfrab.1 . . . . . . 7 𝑥𝜑
87a1i 11 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)
96, 8nfand 1895 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑦𝐴𝜑))
109adantl 481 . . . 4 ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦𝐴𝜑))
112, 10nfabd2 2927 . . 3 (⊤ → 𝑥{𝑦 ∣ (𝑦𝐴𝜑)})
1211mptru 1544 . 2 𝑥{𝑦 ∣ (𝑦𝐴𝜑)}
131, 12nfcxfr 2901 1 𝑥{𝑦𝐴𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wal 1535  wtru 1538  wnf 1780  wcel 2106  {cab 2712  wnfc 2888  {crab 3433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434
This theorem is referenced by:  elfvmptrab1  7044  elovmporab1  7681
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