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Theorem rzal 3731
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 3636 . . . 4  |-  ( x  e.  A  ->  A  =/=  (/) )
21necon2bi 2652 . . 3  |-  ( A  =  (/)  ->  -.  x  e.  A )
32pm2.21d 101 . 2  |-  ( A  =  (/)  ->  ( x  e.  A  ->  ph )
)
43ralrimiv 2790 1  |-  ( A  =  (/)  ->  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   A.wral 2707   (/)c0 3630
This theorem is referenced by:  ralidm  3733  rgenz  3735  ralf0  3736  raaan  3737  raaanv  3738  iinrab2  4156  riinrab  4168  reusv2lem2  4727  cnvpo  5412  dffi3  7438  brdom3  8408  fimaxre2  9958  efgs1  15369  opnnei  17186  ubthlem1  22374  dedekind  25189  nofulllem2  25660  axcontlem12  25916  mbfresfi  26255  bddiblnc  26277  blbnd  26498  rrnequiv  26546  stoweidlem9  27736  raaan2  27931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-v 2960  df-dif 3325  df-nul 3631
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