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Theorem nelrdva 3383
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 2610 . 2 ((𝜑𝐵𝐴) → 𝐵 = 𝐵)
2 eleq1 2675 . . . . . . 7 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
32anbi2d 735 . . . . . 6 (𝑥 = 𝐵 → ((𝜑𝑥𝐴) ↔ (𝜑𝐵𝐴)))
4 neeq1 2843 . . . . . 6 (𝑥 = 𝐵 → (𝑥𝐵𝐵𝐵))
53, 4imbi12d 332 . . . . 5 (𝑥 = 𝐵 → (((𝜑𝑥𝐴) → 𝑥𝐵) ↔ ((𝜑𝐵𝐴) → 𝐵𝐵)))
6 nelrdva.1 . . . . 5 ((𝜑𝑥𝐴) → 𝑥𝐵)
75, 6vtoclg 3238 . . . 4 (𝐵𝐴 → ((𝜑𝐵𝐴) → 𝐵𝐵))
87anabsi7 855 . . 3 ((𝜑𝐵𝐴) → 𝐵𝐵)
98neneqd 2786 . 2 ((𝜑𝐵𝐴) → ¬ 𝐵 = 𝐵)
101, 9pm2.65da 597 1 (𝜑 → ¬ 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wcel 1976  wne 2779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-12 2033  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-ne 2781  df-v 3174
This theorem is referenced by:  ustfilxp  21768  metustfbas  22113  fourierdlem72  38868
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