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Theorem rzal 3375
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 3290 . . . 4 (𝑥𝐴𝐴 ≠ ∅)
21necon2bi 2310 . . 3 (𝐴 = ∅ → ¬ 𝑥𝐴)
32pm2.21d 584 . 2 (𝐴 = ∅ → (𝑥𝐴𝜑))
43ralrimiv 2445 1 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  wcel 1438  wral 2359  c0 3284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-v 2621  df-dif 2999  df-nul 3285
This theorem is referenced by:  ralf0  3381
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