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

Theorem nfraldw 3307
Description: Deduction version of nfralw 3309. Version of nfrald 3370 with a disjoint variable condition, which does not require ax-13 2375. (Contributed by NM, 15-Feb-2013.) Avoid ax-9 2116, ax-ext 2706. (Revised by GG, 24-Sep-2024.)
Hypotheses
Ref Expression
nfraldw.1 𝑦𝜑
nfraldw.2 (𝜑𝑥𝐴)
nfraldw.3 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfraldw (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfraldw
StepHypRef Expression
1 df-ral 3060 . 2 (∀𝑦𝐴 𝜓 ↔ ∀𝑦(𝑦𝐴𝜓))
2 nfraldw.1 . . 3 𝑦𝜑
3 nfraldw.2 . . . . 5 (𝜑𝑥𝐴)
43nfcrd 2897 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐴)
5 nfraldw.3 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
64, 5nfimd 1892 . . 3 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
72, 6nfald 2327 . 2 (𝜑 → Ⅎ𝑥𝑦(𝑦𝐴𝜓))
81, 7nfxfrd 1851 1 (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1535  wnf 1780  wcel 2106  wnfc 2888  wral 3059
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-10 2139  ax-11 2155  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-clel 2814  df-nfc 2890  df-ral 3060
This theorem is referenced by:  nfrexdw  3308  nfralwOLD  3310  nfttrcld  9748
  Copyright terms: Public domain W3C validator