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Theorem ralf0 3318
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 570 . . . . 5 ((x Aφ) → (¬ φ → ¬ x A))
31, 2mpi 15 . . . 4 ((x Aφ) → ¬ x A)
43alimi 1341 . . 3 (x(x Aφ) → x ¬ x A)
5 df-ral 2305 . . 3 (x A φx(x Aφ))
6 eq0 3233 . . 3 (A = ∅ ↔ x ¬ x A)
74, 5, 63imtr4i 190 . 2 (x A φA = ∅)
8 rzal 3312 . 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 1240   = wceq 1242   wcel 1390  wral 2300  c0 3218
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-v 2553  df-dif 2914  df-nul 3219
This theorem is referenced by: (None)
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