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Theorem rzalf 30998
Description: A version of rzal 3929 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 3791 . . . 4  |-  ( x  e.  A  ->  A  =/=  (/) )
32necon2bi 2704 . . 3  |-  ( A  =  (/)  ->  -.  x  e.  A )
43pm2.21d 106 . 2  |-  ( A  =  (/)  ->  ( x  e.  A  ->  ph )
)
51, 4ralrimi 2864 1  |-  ( A  =  (/)  ->  A. x  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379   F/wnf 1599    e. wcel 1767   A.wral 2814   (/)c0 3785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-v 3115  df-dif 3479  df-nul 3786
This theorem is referenced by:  stoweidlem18  31346  stoweidlem28  31356  stoweidlem55  31383
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