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Theorem ralf0 3299
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 (x A φA = ∅)
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem ralf0
StepHypRef Expression
1 ralf0.1 . . . . 5 ¬ φ
2 con3 558 . . . . 5 ((x Aφ) → (¬ φ → ¬ x A))
31, 2mpi 15 . . . 4 ((x Aφ) → ¬ x A)
43alimi 1320 . . 3 (x(x Aφ) → x ¬ x A)
5 df-ral 2285 . . 3 (x A φx(x Aφ))
6 eq0 3212 . . 3 (A = ∅ ↔ x ¬ x A)
74, 5, 63imtr4i 190 . 2 (x A φA = ∅)
8 rzal 3293 . 2 (A = ∅ → x A φ)
97, 8impbii 117 1 (x A φA = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98  wal 1224   = wceq 1226   wcel 1370  wral 2280  c0 3197
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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-v 2533  df-dif 2893  df-nul 3198
This theorem is referenced by: (None)
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