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Theorem rzal 3297
 Description: Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
rzal (A = ∅ → x A φ)
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem rzal
StepHypRef Expression
1 ne0i 3207 . . . 4 (x AA ≠ ∅)
21necon2bi 2238 . . 3 (A = ∅ → ¬ x A)
32pm2.21d 537 . 2 (A = ∅ → (x Aφ))
43ralrimiv 2369 1 (A = ∅ → x A φ)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1228   ∈ wcel 1374  ∀wral 2284  ∅c0 3201 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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-v 2537  df-dif 2897  df-nul 3202 This theorem is referenced by:  ralf0  3303
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