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Theorem rzal 3521
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 = ∅ → ∀𝑥𝐴 𝜑)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rzal
StepHypRef Expression
1 ne0i 3430 . . . 4 (𝑥𝐴𝐴 ≠ ∅)
21necon2bi 2402 . . 3 (𝐴 = ∅ → ¬ 𝑥𝐴)
32pm2.21d 619 . 2 (𝐴 = ∅ → (𝑥𝐴𝜑))
43ralrimiv 2549 1 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  wral 2455  c0 3423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-v 2740  df-dif 3132  df-nul 3424
This theorem is referenced by:  ralf0  3527  fiubm  10808  mgm0  12788  sgrp0  12815
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