MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfrabwOLD Structured version   Visualization version   GIF version

Theorem nfrabwOLD 3440
Description: Obsolete version of nfrabw 3439 as of 23-Nov2024. (Contributed by NM, 13-Oct-2003.) (Revised by Gino Giotto, 10-Jan-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
nfrabw.1 𝑥𝜑
nfrabw.2 𝑥𝐴
Assertion
Ref Expression
nfrabwOLD 𝑥{𝑦𝐴𝜑}
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfrabwOLD
StepHypRef Expression
1 df-rab 3407 . 2 {𝑦𝐴𝜑} = {𝑦 ∣ (𝑦𝐴𝜑)}
2 nftru 1807 . . . 4 𝑦
3 nfrabw.2 . . . . . . 7 𝑥𝐴
43nfcri 2893 . . . . . 6 𝑥 𝑦𝐴
54a1i 11 . . . . 5 (⊤ → Ⅎ𝑥 𝑦𝐴)
6 nfrabw.1 . . . . . 6 𝑥𝜑
76a1i 11 . . . . 5 (⊤ → Ⅎ𝑥𝜑)
85, 7nfand 1901 . . . 4 (⊤ → Ⅎ𝑥(𝑦𝐴𝜑))
92, 8nfabdw 2929 . . 3 (⊤ → 𝑥{𝑦 ∣ (𝑦𝐴𝜑)})
109mptru 1549 . 2 𝑥{𝑦 ∣ (𝑦𝐴𝜑)}
111, 10nfcxfr 2904 1 𝑥{𝑦𝐴𝜑}
Colors of variables: wff setvar class
Syntax hints:  wa 397  wtru 1543  wnf 1786  wcel 2107  {cab 2715  wnfc 2886  {crab 3406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-rab 3407
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator