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

Theorem nelrdva 3384
 Description: Deduce negative membership from an implication. (Contributed by Thierry Arnoux, 27-Nov-2017.)
Hypothesis
Ref Expression
nelrdva.1 ((𝜑𝑥𝐴) → 𝑥𝐵)
Assertion
Ref Expression
nelrdva (𝜑 → ¬ 𝐵𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥

Proof of Theorem nelrdva
StepHypRef Expression
1 eqidd 2611 . 2 ((𝜑𝐵𝐴) → 𝐵 = 𝐵)
2 eleq1 2676 . . . . . . 7 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
32anbi2d 736 . . . . . 6 (𝑥 = 𝐵 → ((𝜑𝑥𝐴) ↔ (𝜑𝐵𝐴)))
4 neeq1 2844 . . . . . 6 (𝑥 = 𝐵 → (𝑥𝐵𝐵𝐵))
53, 4imbi12d 333 . . . . 5 (𝑥 = 𝐵 → (((𝜑𝑥𝐴) → 𝑥𝐵) ↔ ((𝜑𝐵𝐴) → 𝐵𝐵)))
6 nelrdva.1 . . . . 5 ((𝜑𝑥𝐴) → 𝑥𝐵)
75, 6vtoclg 3239 . . . 4 (𝐵𝐴 → ((𝜑𝐵𝐴) → 𝐵𝐵))
87anabsi7 856 . . 3 ((𝜑𝐵𝐴) → 𝐵𝐵)
98neneqd 2787 . 2 ((𝜑𝐵𝐴) → ¬ 𝐵 = 𝐵)
101, 9pm2.65da 598 1 (𝜑 → ¬ 𝐵𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-ne 2782  df-v 3175 This theorem is referenced by:  ustfilxp  21826  metustfbas  22172  fourierdlem72  39071
 Copyright terms: Public domain W3C validator