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Theorem nelrdva 2746
 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 2041 . 2 ((𝜑𝐵𝐴) → 𝐵 = 𝐵)
2 eleq1 2100 . . . . . . 7 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
32anbi2d 437 . . . . . 6 (𝑥 = 𝐵 → ((𝜑𝑥𝐴) ↔ (𝜑𝐵𝐴)))
4 neeq1 2218 . . . . . 6 (𝑥 = 𝐵 → (𝑥𝐵𝐵𝐵))
53, 4imbi12d 223 . . . . 5 (𝑥 = 𝐵 → (((𝜑𝑥𝐴) → 𝑥𝐵) ↔ ((𝜑𝐵𝐴) → 𝐵𝐵)))
6 nelrdva.1 . . . . 5 ((𝜑𝑥𝐴) → 𝑥𝐵)
75, 6vtoclg 2613 . . . 4 (𝐵𝐴 → ((𝜑𝐵𝐴) → 𝐵𝐵))
87anabsi7 515 . . 3 ((𝜑𝐵𝐴) → 𝐵𝐵)
98neneqd 2226 . 2 ((𝜑𝐵𝐴) → ¬ 𝐵 = 𝐵)
101, 9pm2.65da 587 1 (𝜑 → ¬ 𝐵𝐴)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393   ≠ wne 2204 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559 This theorem is referenced by: (None)
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