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Theorem ralf0 3324
 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 571 . . . . 5 ((𝑥𝐴𝜑) → (¬ 𝜑 → ¬ 𝑥𝐴))
31, 2mpi 15 . . . 4 ((𝑥𝐴𝜑) → ¬ 𝑥𝐴)
43alimi 1344 . . 3 (∀𝑥(𝑥𝐴𝜑) → ∀𝑥 ¬ 𝑥𝐴)
5 df-ral 2311 . . 3 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
6 eq0 3239 . . 3 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
74, 5, 63imtr4i 190 . 2 (∀𝑥𝐴 𝜑𝐴 = ∅)
8 rzal 3318 . 2 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
97, 8impbii 117 1 (∀𝑥𝐴 𝜑𝐴 = ∅)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98  ∀wal 1241   = wceq 1243   ∈ wcel 1393  ∀wral 2306  ∅c0 3224 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-v 2559  df-dif 2920  df-nul 3225 This theorem is referenced by: (None)
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