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Theorem nelrdva 3695
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 2822 . 2 ((𝜑𝐵𝐴) → 𝐵 = 𝐵)
2 eleq1 2900 . . . . . . 7 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
32anbi2d 628 . . . . . 6 (𝑥 = 𝐵 → ((𝜑𝑥𝐴) ↔ (𝜑𝐵𝐴)))
4 neeq1 3078 . . . . . 6 (𝑥 = 𝐵 → (𝑥𝐵𝐵𝐵))
53, 4imbi12d 346 . . . . 5 (𝑥 = 𝐵 → (((𝜑𝑥𝐴) → 𝑥𝐵) ↔ ((𝜑𝐵𝐴) → 𝐵𝐵)))
6 nelrdva.1 . . . . 5 ((𝜑𝑥𝐴) → 𝑥𝐵)
75, 6vtoclg 3568 . . . 4 (𝐵𝐴 → ((𝜑𝐵𝐴) → 𝐵𝐵))
87anabsi7 667 . . 3 ((𝜑𝐵𝐴) → 𝐵𝐵)
98neneqd 3021 . 2 ((𝜑𝐵𝐴) → ¬ 𝐵 = 𝐵)
101, 9pm2.65da 813 1 (𝜑 → ¬ 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  wne 3016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-12 2167  ax-ext 2793
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-nf 1776  df-cleq 2814  df-clel 2893  df-ne 3017
This theorem is referenced by:  ustfilxp  22750  metustfbas  23096  fourierdlem72  42344
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