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Theorem rzal 3466
 Description: Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
rzal (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rzal
StepHypRef Expression
1 ne0i 3375 . . . 4 (𝑥𝐴𝐴 ≠ ∅)
21necon2bi 2364 . . 3 (𝐴 = ∅ → ¬ 𝑥𝐴)
32pm2.21d 609 . 2 (𝐴 = ∅ → (𝑥𝐴𝜑))
43ralrimiv 2508 1 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ∅c0 3369 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-v 2692  df-dif 3079  df-nul 3370 This theorem is referenced by:  ralf0  3472
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