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Theorem ralf0 3352
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 604 . . . . 5 ((𝑥𝐴𝜑) → (¬ 𝜑 → ¬ 𝑥𝐴))
31, 2mpi 15 . . . 4 ((𝑥𝐴𝜑) → ¬ 𝑥𝐴)
43alimi 1385 . . 3 (∀𝑥(𝑥𝐴𝜑) → ∀𝑥 ¬ 𝑥𝐴)
5 df-ral 2354 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
6 eq0 3273 . . 3 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
74, 5, 63imtr4i 199 . 2 (∀𝑥𝐴 𝜑𝐴 = ∅)
8 rzal 3346 . 2 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
97, 8impbii 124 1 (∀𝑥𝐴 𝜑𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103  wal 1283   = wceq 1285  wcel 1434  wral 2349  c0 3258
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-v 2604  df-dif 2976  df-nul 3259
This theorem is referenced by: (None)
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