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Theorem nelrdva 2740
Description: Deduce negative membership from an implication. (Contributed by Thierry Arnoux, 27-Nov-2017.)
Hypothesis
Ref Expression
nelrdva.1 ((φ x A) → xB)
Assertion
Ref Expression
nelrdva (φ → ¬ B A)
Distinct variable groups:   x,A   x,B   φ,x

Proof of Theorem nelrdva
StepHypRef Expression
1 eqidd 2038 . 2 ((φ B A) → B = B)
2 eleq1 2097 . . . . . . 7 (x = B → (x AB A))
32anbi2d 437 . . . . . 6 (x = B → ((φ x A) ↔ (φ B A)))
4 neeq1 2213 . . . . . 6 (x = B → (xBBB))
53, 4imbi12d 223 . . . . 5 (x = B → (((φ x A) → xB) ↔ ((φ B A) → BB)))
6 nelrdva.1 . . . . 5 ((φ x A) → xB)
75, 6vtoclg 2607 . . . 4 (B A → ((φ B A) → BB))
87anabsi7 515 . . 3 ((φ B A) → BB)
98neneqd 2221 . 2 ((φ B A) → ¬ B = B)
101, 9pm2.65da 586 1 (φ → ¬ B A)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   = wceq 1242   wcel 1390  wne 2201
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-v 2553
This theorem is referenced by: (None)
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