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

Theorem nfneld 3131
 Description: Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypotheses
Ref Expression
nfneld.1 (𝜑𝑥𝐴)
nfneld.2 (𝜑𝑥𝐵)
Assertion
Ref Expression
nfneld (𝜑 → Ⅎ𝑥 𝐴𝐵)

Proof of Theorem nfneld
StepHypRef Expression
1 df-nel 3124 . 2 (𝐴𝐵 ↔ ¬ 𝐴𝐵)
2 nfneld.1 . . . 4 (𝜑𝑥𝐴)
3 nfneld.2 . . . 4 (𝜑𝑥𝐵)
42, 3nfeld 2989 . . 3 (𝜑 → Ⅎ𝑥 𝐴𝐵)
54nfnd 1854 . 2 (𝜑 → Ⅎ𝑥 ¬ 𝐴𝐵)
61, 5nfxfrd 1850 1 (𝜑 → Ⅎ𝑥 𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  Ⅎwnf 1780   ∈ wcel 2110  Ⅎwnfc 2961   ∉ wnel 3123 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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-nf 1781  df-cleq 2814  df-clel 2893  df-nfc 2963  df-nel 3124 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator