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Theorem nelrdva 3450
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 2652 . 2 ((𝜑𝐵𝐴) → 𝐵 = 𝐵)
2 eleq1 2718 . . . . . . 7 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
32anbi2d 740 . . . . . 6 (𝑥 = 𝐵 → ((𝜑𝑥𝐴) ↔ (𝜑𝐵𝐴)))
4 neeq1 2885 . . . . . 6 (𝑥 = 𝐵 → (𝑥𝐵𝐵𝐵))
53, 4imbi12d 333 . . . . 5 (𝑥 = 𝐵 → (((𝜑𝑥𝐴) → 𝑥𝐵) ↔ ((𝜑𝐵𝐴) → 𝐵𝐵)))
6 nelrdva.1 . . . . 5 ((𝜑𝑥𝐴) → 𝑥𝐵)
75, 6vtoclg 3297 . . . 4 (𝐵𝐴 → ((𝜑𝐵𝐴) → 𝐵𝐵))
87anabsi7 877 . . 3 ((𝜑𝐵𝐴) → 𝐵𝐵)
98neneqd 2828 . 2 ((𝜑𝐵𝐴) → ¬ 𝐵 = 𝐵)
101, 9pm2.65da 599 1 (𝜑 → ¬ 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  wne 2823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-12 2087  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-ne 2824  df-v 3233
This theorem is referenced by:  ustfilxp  22063  metustfbas  22409  fourierdlem72  40713
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