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Theorem rzalf 27655
Description: A version of rzal 3721 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 3626 . . . 4  |-  ( x  e.  A  ->  A  =/=  (/) )
32necon2bi 2644 . . 3  |-  ( A  =  (/)  ->  -.  x  e.  A )
43pm2.21d 100 . 2  |-  ( A  =  (/)  ->  ( x  e.  A  ->  ph )
)
51, 4ralrimi 2779 1  |-  ( A  =  (/)  ->  A. x  e.  A  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1553    = wceq 1652    e. wcel 1725   A.wral 2697   (/)c0 3620
This theorem is referenced by:  stoweidlem18  27734  stoweidlem28  27744  stoweidlem55  27771
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-v 2950  df-dif 3315  df-nul 3621
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