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Theorem rzal 3589
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  =  (/)  ->  A. x  e.  A  ph )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem rzal
StepHypRef Expression
1 ne0i 3495 . . . 4  |-  ( x  e.  A  ->  A  =/=  (/) )
21necon2bi 2525 . . 3  |-  ( A  =  (/)  ->  -.  x  e.  A )
32pm2.21d 98 . 2  |-  ( A  =  (/)  ->  ( x  e.  A  ->  ph )
)
43ralrimiv 2659 1  |-  ( A  =  (/)  ->  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1633    e. wcel 1701   A.wral 2577   (/)c0 3489
This theorem is referenced by:  ralidm  3591  rgenz  3593  ralf0  3594  raaan  3595  raaanv  3596  iinrab2  4002  riinrab  4014  reusv2lem2  4573  cnvpo  5250  dffi3  7229  brdom3  8198  fimaxre2  9747  efgs1  15093  opnnei  16913  ubthlem1  21504  dedekind  24368  nofulllem2  24742  axcontlem12  24989  bddiblnc  25335  blbnd  25659  rrnequiv  25707  stoweidlem9  26906  raaan2  27101
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-v 2824  df-dif 3189  df-nul 3490
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