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Theorem rzalf 27688
Description: A version of rzal 3555 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 3461 . . . 4  |-  ( x  e.  A  ->  A  =/=  (/) )
32necon2bi 2492 . . 3  |-  ( A  =  (/)  ->  -.  x  e.  A )
43pm2.21d 98 . 2  |-  ( A  =  (/)  ->  ( x  e.  A  ->  ph )
)
51, 4ralrimi 2624 1  |-  ( A  =  (/)  ->  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1531    = wceq 1623    e. wcel 1684   A.wral 2543   (/)c0 3455
This theorem is referenced by:  stoweidlem18  27767  stoweidlem28  27777  stoweidlem55  27804
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-v 2790  df-dif 3155  df-nul 3456
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