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Theorem ralf0 3413
Description: The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)
Hypothesis
Ref Expression
ralf0.1 ¬ 𝜑
Assertion
Ref Expression
ralf0 (∀𝑥𝐴 𝜑𝐴 = ∅)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ralf0
StepHypRef Expression
1 ralf0.1 . . . . 5 ¬ 𝜑
2 con3 611 . . . . 5 ((𝑥𝐴𝜑) → (¬ 𝜑 → ¬ 𝑥𝐴))
31, 2mpi 15 . . . 4 ((𝑥𝐴𝜑) → ¬ 𝑥𝐴)
43alimi 1399 . . 3 (∀𝑥(𝑥𝐴𝜑) → ∀𝑥 ¬ 𝑥𝐴)
5 df-ral 2380 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
6 eq0 3328 . . 3 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
74, 5, 63imtr4i 200 . 2 (∀𝑥𝐴 𝜑𝐴 = ∅)
8 rzal 3407 . 2 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
97, 8impbii 125 1 (∀𝑥𝐴 𝜑𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104  wal 1297   = wceq 1299  wcel 1448  wral 2375  c0 3310
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-v 2643  df-dif 3023  df-nul 3311
This theorem is referenced by: (None)
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