Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rzalf Structured version   Unicode version

Theorem rzalf 36978
Description: A version of rzal 3905 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypothesis
Ref Expression
rzalf.1  |-  F/ x  A  =  (/)
Assertion
Ref Expression
rzalf  |-  ( A  =  (/)  ->  A. x  e.  A  ph )

Proof of Theorem rzalf
StepHypRef Expression
1 rzalf.1 . 2  |-  F/ x  A  =  (/)
2 ne0i 3773 . . . 4  |-  ( x  e.  A  ->  A  =/=  (/) )
32necon2bi 2668 . . 3  |-  ( A  =  (/)  ->  -.  x  e.  A )
43pm2.21d 109 . 2  |-  ( A  =  (/)  ->  ( x  e.  A  ->  ph )
)
51, 4ralrimi 2832 1  |-  ( A  =  (/)  ->  A. x  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437   F/wnf 1663    e. wcel 1870   A.wral 2782   (/)c0 3767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-v 3089  df-dif 3445  df-nul 3768
This theorem is referenced by:  stoweidlem18  37447  stoweidlem28  37457  stoweidlem55  37485
  Copyright terms: Public domain W3C validator